| L(s) = 1 | − 16·7-s − 8·13-s − 40·19-s + 44·25-s + 56·31-s + 136·37-s + 104·43-s − 116·49-s − 8·61-s − 112·67-s + 448·73-s − 448·79-s + 128·91-s − 152·97-s + 104·103-s − 680·109-s − 44·121-s + 127-s + 131-s + 640·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.28·7-s − 0.615·13-s − 2.10·19-s + 1.75·25-s + 1.80·31-s + 3.67·37-s + 2.41·43-s − 2.36·49-s − 0.131·61-s − 1.67·67-s + 6.13·73-s − 5.67·79-s + 1.40·91-s − 1.56·97-s + 1.00·103-s − 6.23·109-s − 0.363·121-s + 0.00787·127-s + 0.00763·131-s + 4.81·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4923650090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4923650090\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( ( 1 + p T^{2} )^{4} \) |
| good | 5 | \( 1 - 44 T^{2} + 2896 T^{4} - 15972 p T^{6} + 2805486 T^{8} - 15972 p^{5} T^{10} + 2896 p^{8} T^{12} - 44 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 8 T + 22 p T^{2} + 160 p T^{3} + 10526 T^{4} + 160 p^{3} T^{5} + 22 p^{5} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4 T + 22 p T^{2} - 388 T^{3} + 42374 T^{4} - 388 p^{2} T^{5} + 22 p^{5} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1232 T^{2} + 735100 T^{4} - 289134384 T^{6} + 90323356422 T^{8} - 289134384 p^{4} T^{10} + 735100 p^{8} T^{12} - 1232 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 + 20 T + 868 T^{2} + 8916 T^{3} + 328146 T^{4} + 8916 p^{2} T^{5} + 868 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 2444 T^{2} + 3254416 T^{4} - 2852397780 T^{6} + 1774428987822 T^{8} - 2852397780 p^{4} T^{10} + 3254416 p^{8} T^{12} - 2444 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 1880 T^{2} + 1924252 T^{4} - 2074330856 T^{6} + 2128273704070 T^{8} - 2074330856 p^{4} T^{10} + 1924252 p^{8} T^{12} - 1880 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 28 T + 1432 T^{2} - 13412 T^{3} + 1149278 T^{4} - 13412 p^{2} T^{5} + 1432 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 68 T + 4504 T^{2} - 191964 T^{3} + 7669374 T^{4} - 191964 p^{2} T^{5} + 4504 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 8624 T^{2} + 37064956 T^{4} - 103569069648 T^{6} + 204770887955334 T^{8} - 103569069648 p^{4} T^{10} + 37064956 p^{8} T^{12} - 8624 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 52 T + 3460 T^{2} - 22452 T^{3} + 2716146 T^{4} - 22452 p^{2} T^{5} + 3460 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 6620 T^{2} + 27373072 T^{4} - 80988531396 T^{6} + 198687369383598 T^{8} - 80988531396 p^{4} T^{10} + 27373072 p^{8} T^{12} - 6620 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 14348 T^{2} + 100839952 T^{4} - 466253289236 T^{6} + 1539595638444910 T^{8} - 466253289236 p^{4} T^{10} + 100839952 p^{8} T^{12} - 14348 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 19976 T^{2} + 192056668 T^{4} - 1159676388024 T^{6} + 4810763889113478 T^{8} - 1159676388024 p^{4} T^{10} + 192056668 p^{8} T^{12} - 19976 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 4 T + 8686 T^{2} + 150828 T^{3} + 38662374 T^{4} + 150828 p^{2} T^{5} + 8686 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 56 T + 15436 T^{2} + 560264 T^{3} + 95435590 T^{4} + 560264 p^{2} T^{5} + 15436 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 22412 T^{2} + 258098896 T^{4} - 2017610554196 T^{6} + 11687868067441198 T^{8} - 2017610554196 p^{4} T^{10} + 258098896 p^{8} T^{12} - 22412 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 224 T + 28780 T^{2} - 2677024 T^{3} + 212472614 T^{4} - 2677024 p^{2} T^{5} + 28780 p^{4} T^{6} - 224 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 224 T + 40282 T^{2} + 4437320 T^{3} + 5320162 p T^{4} + 4437320 p^{2} T^{5} + 40282 p^{4} T^{6} + 224 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 26528 T^{2} + 386566684 T^{4} - 3936191194208 T^{6} + 30559983176028934 T^{8} - 3936191194208 p^{4} T^{10} + 386566684 p^{8} T^{12} - 26528 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 13664 T^{2} + 265530628 T^{4} - 2460680918048 T^{6} + 25256672766991750 T^{8} - 2460680918048 p^{4} T^{10} + 265530628 p^{8} T^{12} - 13664 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 76 T + 22576 T^{2} + 814804 T^{3} + 238699630 T^{4} + 814804 p^{2} T^{5} + 22576 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.65829819771753825075953044784, −3.63194938419041573832507175221, −3.50512259947264410110579554373, −3.34902549949287365991359387023, −3.25284768066112492628288316044, −3.22521764326752071163056318266, −2.81197935971472881563430354570, −2.80833909071347147290039768949, −2.62206902363993402314182422497, −2.59037259014859337208082298104, −2.54968988954370095925314881741, −2.50791107135214422548089288100, −2.40943484420917676160266559641, −2.22565777960903220277608826347, −1.80383397047808155865281243171, −1.58429762556517384645368579402, −1.56006321788222548119611289920, −1.44832822182842890221239541077, −1.19322805217005766348619454151, −0.981337021087652131914697979169, −0.941825803495554317333389207388, −0.73565631840000546129069396716, −0.29866670447237979631722448516, −0.29563649342085073276951884752, −0.092460964581285303012100652718,
0.092460964581285303012100652718, 0.29563649342085073276951884752, 0.29866670447237979631722448516, 0.73565631840000546129069396716, 0.941825803495554317333389207388, 0.981337021087652131914697979169, 1.19322805217005766348619454151, 1.44832822182842890221239541077, 1.56006321788222548119611289920, 1.58429762556517384645368579402, 1.80383397047808155865281243171, 2.22565777960903220277608826347, 2.40943484420917676160266559641, 2.50791107135214422548089288100, 2.54968988954370095925314881741, 2.59037259014859337208082298104, 2.62206902363993402314182422497, 2.80833909071347147290039768949, 2.81197935971472881563430354570, 3.22521764326752071163056318266, 3.25284768066112492628288316044, 3.34902549949287365991359387023, 3.50512259947264410110579554373, 3.63194938419041573832507175221, 3.65829819771753825075953044784
Plot not available for L-functions of degree greater than 10.
