| L(s) = 1 | + 16·9-s − 24·11-s − 68·19-s − 400·49-s − 600·61-s + 80·81-s − 384·99-s − 424·101-s − 940·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 664·169-s − 1.08e3·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 16/9·9-s − 2.18·11-s − 3.57·19-s − 8.16·49-s − 9.83·61-s + 0.987·81-s − 3.87·99-s − 4.19·101-s − 7.76·121-s + 0.00787·127-s + 0.00763·131-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 + 3.92·169-s − 6.36·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3768772772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3768772772\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + 34 T + 723 T^{2} + 636 p T^{3} + 723 p^{2} T^{4} + 34 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| good | 3 | \( ( 1 - 8 T^{2} + 56 T^{4} + 110 T^{6} + 56 p^{4} T^{8} - 8 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 7 | \( ( 1 + 200 T^{2} + 19984 T^{4} + 24978 p^{2} T^{6} + 19984 p^{4} T^{8} + 200 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( ( 1 + 6 T + 325 T^{2} + 128 p T^{3} + 325 p^{2} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 13 | \( ( 1 - 332 T^{2} + 7272 p T^{4} - 15442282 T^{6} + 7272 p^{5} T^{8} - 332 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 17 | \( ( 1 + 1420 T^{2} + 900624 T^{4} + 331623882 T^{6} + 900624 p^{4} T^{8} + 1420 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 23 | \( ( 1 + 824 T^{2} + 720640 T^{4} + 450330930 T^{6} + 720640 p^{4} T^{8} + 824 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 29 | \( ( 1 - 2740 T^{2} + 4126128 T^{4} - 4996698 p^{2} T^{6} + 4126128 p^{4} T^{8} - 2740 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 31 | \( ( 1 - 1114 T^{2} + 641475 T^{4} + 395464236 T^{6} + 641475 p^{4} T^{8} - 1114 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 37 | \( ( 1 - 2902 T^{2} + 8216351 T^{4} - 11547943732 T^{6} + 8216351 p^{4} T^{8} - 2902 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 41 | \( ( 1 - 2946 T^{2} - 2428413 T^{4} + 15632004796 T^{6} - 2428413 p^{4} T^{8} - 2946 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 43 | \( ( 1 + 3554 T^{2} + 8177587 T^{4} + 18487447044 T^{6} + 8177587 p^{4} T^{8} + 3554 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 47 | \( ( 1 - 802 T^{2} + 9915583 T^{4} - 2414107836 T^{6} + 9915583 p^{4} T^{8} - 802 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 53 | \( ( 1 - 13724 T^{2} + 85490808 T^{4} - 307901289994 T^{6} + 85490808 p^{4} T^{8} - 13724 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 59 | \( ( 1 - 2800 T^{2} + 12497808 T^{4} - 96285230418 T^{6} + 12497808 p^{4} T^{8} - 2800 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 61 | \( ( 1 + 150 T + 16213 T^{2} + 1135264 T^{3} + 16213 p^{2} T^{4} + 150 p^{4} T^{5} + p^{6} T^{6} )^{4} \) |
| 67 | \( ( 1 - 4504 T^{2} + 47039432 T^{4} - 119843552146 T^{6} + 47039432 p^{4} T^{8} - 4504 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 71 | \( ( 1 - 18266 T^{2} + 170784163 T^{4} - 1026393192276 T^{6} + 170784163 p^{4} T^{8} - 18266 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 73 | \( ( 1 + 19260 T^{2} + 179069184 T^{4} + 1112885983082 T^{6} + 179069184 p^{4} T^{8} + 19260 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 79 | \( ( 1 - 11006 T^{2} + 39589567 T^{4} - 118680581124 T^{6} + 39589567 p^{4} T^{8} - 11006 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 83 | \( ( 1 + 21718 T^{2} + 291339135 T^{4} + 2389791397812 T^{6} + 291339135 p^{4} T^{8} + 21718 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 89 | \( ( 1 - 29018 T^{2} + 5189339 p T^{4} - 4481569968468 T^{6} + 5189339 p^{5} T^{8} - 29018 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 97 | \( ( 1 - 34446 T^{2} + 650399935 T^{4} - 7487480008036 T^{6} + 650399935 p^{4} T^{8} - 34446 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
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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
−2.70640427071815480877158489621, −2.49701859030550024407408852040, −2.45612805456946558020349193832, −2.44354807429714200617676007888, −2.27699923536294836578271579918, −2.13959256564197008690248040349, −2.04228749795482789647504606258, −1.96571668935278188111139857162, −1.77879590956963485126541972100, −1.68195950895413588574088757002, −1.64770732531523177850019671881, −1.61549287917761063573478997258, −1.52248615125040244008062970666, −1.41854694095475936872972979017, −1.41790005864460272246218372347, −1.26962603462727380252331800406, −1.25055235908336167298562109090, −1.08717804670119143486319717827, −0.76868292518209222117309098626, −0.56574695894402121285468394031, −0.43650133033859319990376703056, −0.32512441430490456808850636979, −0.21270206689971456339766740622, −0.15381977335984468860711131791, −0.10063367829220773898271280290,
0.10063367829220773898271280290, 0.15381977335984468860711131791, 0.21270206689971456339766740622, 0.32512441430490456808850636979, 0.43650133033859319990376703056, 0.56574695894402121285468394031, 0.76868292518209222117309098626, 1.08717804670119143486319717827, 1.25055235908336167298562109090, 1.26962603462727380252331800406, 1.41790005864460272246218372347, 1.41854694095475936872972979017, 1.52248615125040244008062970666, 1.61549287917761063573478997258, 1.64770732531523177850019671881, 1.68195950895413588574088757002, 1.77879590956963485126541972100, 1.96571668935278188111139857162, 2.04228749795482789647504606258, 2.13959256564197008690248040349, 2.27699923536294836578271579918, 2.44354807429714200617676007888, 2.45612805456946558020349193832, 2.49701859030550024407408852040, 2.70640427071815480877158489621
Plot not available for L-functions of degree greater than 10.
