| L(s) = 1 | − 24·13-s − 16·37-s + 44·49-s − 8·61-s − 104·73-s − 40·97-s − 32·109-s − 76·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 196·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 6.65·13-s − 2.63·37-s + 44/7·49-s − 1.02·61-s − 12.1·73-s − 4.06·97-s − 3.06·109-s − 6.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 15.0·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05823927498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05823927498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 22 T^{2} + 275 T^{4} - 2316 T^{6} + 275 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 38 T^{2} + 811 T^{4} + 10796 T^{6} + 811 p^{2} T^{8} + 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 + 6 T + 41 T^{2} + 152 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 17 | \( ( 1 - 4 p T^{2} + 2311 T^{4} - 48968 T^{6} + 2311 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 34 T^{2} + 935 T^{4} - 20604 T^{6} + 935 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 + 50 T^{2} + 367 T^{4} - 12196 T^{6} + 367 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 76 T^{2} + 3391 T^{4} - 114424 T^{6} + 3391 p^{2} T^{8} - 76 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 98 T^{2} + 4927 T^{4} - 170300 T^{6} + 4927 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 4 T + 97 T^{2} + 304 T^{3} + 97 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 144 T^{2} + 11315 T^{4} - 13856 p T^{6} + 11315 p^{2} T^{8} - 144 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 18 T^{2} + 2903 T^{4} - 79964 T^{6} + 2903 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 106 T^{2} + 5743 T^{4} + 227980 T^{6} + 5743 p^{2} T^{8} + 106 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 12 T^{2} + 7727 T^{4} - 58168 T^{6} + 7727 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 134 T^{2} + 15947 T^{4} + 1000044 T^{6} + 15947 p^{2} T^{8} + 134 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 2 T + 83 T^{2} - 84 T^{3} + 83 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 - 178 T^{2} + 21895 T^{4} - 1675228 T^{6} + 21895 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 170 T^{2} + 18527 T^{4} + 1427916 T^{6} + 18527 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 26 T + 419 T^{2} + 4260 T^{3} + 419 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 79 | \( ( 1 - 130 T^{2} + 14495 T^{4} - 1438332 T^{6} + 14495 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 242 T^{2} + 33959 T^{4} + 3241692 T^{6} + 33959 p^{2} T^{8} + 242 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 416 T^{2} + 78163 T^{4} - 8732480 T^{6} + 78163 p^{2} T^{8} - 416 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 10 T + 203 T^{2} + 1092 T^{3} + 203 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
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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
−2.31929056906471431793689850737, −2.25561138067819177244454184119, −2.23177294681288996741266411928, −2.15943548609931364284889718006, −2.14619646911895940736399336383, −2.07798531869796505332165328258, −2.07520665392735401027244719427, −1.72064705859625220526321266561, −1.57768942228682337509732065066, −1.50953266960931074906199370426, −1.42835077855263843433456643493, −1.40041818244550648008951895795, −1.34643018950370257168363291079, −1.31343615115205320346303523365, −1.17547664053233516491093292240, −1.14164872968098961711675443475, −1.04012547736529271674059685524, −0.968147040408770659849810108576, −0.890829345714030101716830300944, −0.47447951100833087593107509687, −0.30011146232006298849691846164, −0.21620680389475626313847420385, −0.18806644683990141813447373595, −0.11500156937061728874698343851, −0.10965882105940201223213821020,
0.10965882105940201223213821020, 0.11500156937061728874698343851, 0.18806644683990141813447373595, 0.21620680389475626313847420385, 0.30011146232006298849691846164, 0.47447951100833087593107509687, 0.890829345714030101716830300944, 0.968147040408770659849810108576, 1.04012547736529271674059685524, 1.14164872968098961711675443475, 1.17547664053233516491093292240, 1.31343615115205320346303523365, 1.34643018950370257168363291079, 1.40041818244550648008951895795, 1.42835077855263843433456643493, 1.50953266960931074906199370426, 1.57768942228682337509732065066, 1.72064705859625220526321266561, 2.07520665392735401027244719427, 2.07798531869796505332165328258, 2.14619646911895940736399336383, 2.15943548609931364284889718006, 2.23177294681288996741266411928, 2.25561138067819177244454184119, 2.31929056906471431793689850737
Plot not available for L-functions of degree greater than 10.
