| L(s) = 1 | − 60·5-s + 620·9-s + 2.55e3·25-s − 2.13e4·29-s − 1.09e5·41-s − 3.72e4·45-s + 8.46e4·49-s + 6.45e4·61-s + 4.69e4·81-s − 3.72e3·89-s + 3.05e5·101-s + 7.62e5·109-s − 4.40e5·121-s + 1.52e5·125-s + 127-s + 131-s + 137-s + 139-s + 1.27e6·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.84e6·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + 2.55·9-s + 0.815·25-s − 4.70·29-s − 10.1·41-s − 2.73·45-s + 5.03·49-s + 2.22·61-s + 0.794·81-s − 0.0497·89-s + 2.98·101-s + 6.14·109-s − 2.73·121-s + 0.872·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 5.04·145-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4.96·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2686274947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2686274947\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + 6 p T + 3 p^{2} T^{2} - 988 p^{3} T^{3} + 3 p^{7} T^{4} + 6 p^{11} T^{5} + p^{15} T^{6} )^{2} \) |
| good | 3 | \( ( 1 - 310 T^{2} + 120679 T^{4} - 8899676 p T^{6} + 120679 p^{10} T^{8} - 310 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 7 | \( ( 1 - 42334 T^{2} + 106904873 p T^{4} - 10497411086308 T^{6} + 106904873 p^{11} T^{8} - 42334 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 11 | \( ( 1 + 220194 T^{2} + 46554422967 T^{4} + 4986677846505148 T^{6} + 46554422967 p^{10} T^{8} + 220194 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 13 | \( ( 1 - 921822 T^{2} + 378030377607 T^{4} - 125995996576493956 T^{6} + 378030377607 p^{10} T^{8} - 921822 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 17 | \( ( 1 - 441670 p T^{2} + 24616526523247 T^{4} - 45351621021996027220 T^{6} + 24616526523247 p^{10} T^{8} - 441670 p^{21} T^{10} + p^{30} T^{12} )^{2} \) |
| 19 | \( ( 1 + 8862994 T^{2} + 40251921150215 T^{4} + \)\(12\!\cdots\!80\)\( T^{6} + 40251921150215 p^{10} T^{8} + 8862994 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 23 | \( ( 1 - 27906430 T^{2} + 367961637958719 T^{4} - \)\(29\!\cdots\!08\)\( T^{6} + 367961637958719 p^{10} T^{8} - 27906430 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 29 | \( ( 1 + 5326 T + 46243827 T^{2} + 133362268948 T^{3} + 46243827 p^{5} T^{4} + 5326 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 31 | \( ( 1 + 10594234 T^{2} - 153174449706673 T^{4} + \)\(19\!\cdots\!88\)\( T^{6} - 153174449706673 p^{10} T^{8} + 10594234 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 37 | \( ( 1 - 234974254 T^{2} + 32080895329334807 T^{4} - \)\(26\!\cdots\!92\)\( T^{6} + 32080895329334807 p^{10} T^{8} - 234974254 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 41 | \( ( 1 + 27418 T + 531273543 T^{2} + 6402679071436 T^{3} + 531273543 p^{5} T^{4} + 27418 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 43 | \( ( 1 - 613900486 T^{2} + 182904754783784951 T^{4} - \)\(33\!\cdots\!72\)\( T^{6} + 182904754783784951 p^{10} T^{8} - 613900486 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 47 | \( ( 1 - 579920494 T^{2} + 165136673035562991 T^{4} - \)\(38\!\cdots\!48\)\( T^{6} + 165136673035562991 p^{10} T^{8} - 579920494 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 53 | \( ( 1 - 2026325390 T^{2} + 1866514545072595447 T^{4} - \)\(99\!\cdots\!20\)\( T^{6} + 1866514545072595447 p^{10} T^{8} - 2026325390 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 59 | \( ( 1 + 55950786 T^{2} + 253972763776609047 T^{4} - \)\(45\!\cdots\!68\)\( T^{6} + 253972763776609047 p^{10} T^{8} + 55950786 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 61 | \( ( 1 - 16138 T + 744681843 T^{2} - 29097452426876 T^{3} + 744681843 p^{5} T^{4} - 16138 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 67 | \( ( 1 - 2173336374 T^{2} + 6882868324909910631 T^{4} - \)\(82\!\cdots\!68\)\( T^{6} + 6882868324909910631 p^{10} T^{8} - 2173336374 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 71 | \( ( 1 + 942628714 T^{2} + 4622069102042239647 T^{4} + \)\(48\!\cdots\!68\)\( T^{6} + 4622069102042239647 p^{10} T^{8} + 942628714 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 73 | \( ( 1 - 4736672630 T^{2} + 15959781190221205247 T^{4} - \)\(37\!\cdots\!40\)\( T^{6} + 15959781190221205247 p^{10} T^{8} - 4736672630 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 79 | \( ( 1 + 454164826 T^{2} + 18487132928554429487 T^{4} - \)\(50\!\cdots\!28\)\( T^{6} + 18487132928554429487 p^{10} T^{8} + 454164826 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 83 | \( ( 1 - 11599602390 T^{2} + 72197091204656028039 T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + 72197091204656028039 p^{10} T^{8} - 11599602390 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
| 89 | \( ( 1 + 930 T + 9446079447 T^{2} + 231471892494140 T^{3} + 9446079447 p^{5} T^{4} + 930 p^{10} T^{5} + p^{15} T^{6} )^{4} \) |
| 97 | \( ( 1 - 35609466630 T^{2} + \)\(64\!\cdots\!47\)\( T^{4} - \)\(68\!\cdots\!40\)\( T^{6} + \)\(64\!\cdots\!47\)\( p^{10} T^{8} - 35609466630 p^{20} T^{10} + p^{30} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.58670910377359614415573878227, −3.49653884561178876254714808338, −3.37724801904333355202847118174, −3.37277203347905508417403095132, −3.20009746724772911626633911514, −2.81974946302246736903492558203, −2.64018780071023589719072794916, −2.57080955443150374792066600907, −2.46151803208059319307785816716, −2.28939242844623605100260092962, −2.00408061202463107389115906174, −1.97924180087885245247115442740, −1.90777925296490515017176345246, −1.74583305100143525204530096768, −1.67321962541568770451239041793, −1.58231821421025770857710110179, −1.41428270642166964117653125424, −1.22077940499381805559895143774, −0.988460008103250844748982928581, −0.820834974234795127537759590283, −0.77311123615663630385380667180, −0.63932452637228997311136350731, −0.24975165804457421290051697008, −0.13357109662032687206075867166, −0.07428407952765529821511217804,
0.07428407952765529821511217804, 0.13357109662032687206075867166, 0.24975165804457421290051697008, 0.63932452637228997311136350731, 0.77311123615663630385380667180, 0.820834974234795127537759590283, 0.988460008103250844748982928581, 1.22077940499381805559895143774, 1.41428270642166964117653125424, 1.58231821421025770857710110179, 1.67321962541568770451239041793, 1.74583305100143525204530096768, 1.90777925296490515017176345246, 1.97924180087885245247115442740, 2.00408061202463107389115906174, 2.28939242844623605100260092962, 2.46151803208059319307785816716, 2.57080955443150374792066600907, 2.64018780071023589719072794916, 2.81974946302246736903492558203, 3.20009746724772911626633911514, 3.37277203347905508417403095132, 3.37724801904333355202847118174, 3.49653884561178876254714808338, 3.58670910377359614415573878227
Plot not available for L-functions of degree greater than 10.
