| L(s) = 1 | + 2.18e3·3-s − 1.14e5·5-s + 3.03e6·7-s + 4.78e6·9-s + 1.03e8·11-s − 1.04e8·13-s − 2.51e8·15-s + 9.97e8·17-s − 4.93e9·19-s + 6.63e9·21-s − 8.32e9·23-s − 1.73e10·25-s + 1.04e10·27-s + 1.04e11·29-s + 2.96e11·31-s + 2.26e11·33-s − 3.48e11·35-s − 1.78e11·37-s − 2.28e11·39-s − 1.79e12·41-s + 2.86e12·43-s − 5.49e11·45-s − 4.33e12·47-s + 4.46e12·49-s + 2.18e12·51-s + 9.73e12·53-s − 1.18e13·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.657·5-s + 1.39·7-s + 1/3·9-s + 1.60·11-s − 0.461·13-s − 0.379·15-s + 0.589·17-s − 1.26·19-s + 0.804·21-s − 0.509·23-s − 0.568·25-s + 0.192·27-s + 1.12·29-s + 1.93·31-s + 0.924·33-s − 0.915·35-s − 0.308·37-s − 0.266·39-s − 1.43·41-s + 1.60·43-s − 0.219·45-s − 1.24·47-s + 0.939·49-s + 0.340·51-s + 1.13·53-s − 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(3.137282743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.137282743\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{7} T \) |
| good | 5 | \( 1 + 22962 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 433504 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 9404700 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 104365834 T + p^{15} T^{2} \) |
| 17 | \( 1 - 997689762 T + p^{15} T^{2} \) |
| 19 | \( 1 + 4934015444 T + p^{15} T^{2} \) |
| 23 | \( 1 + 8324920200 T + p^{15} T^{2} \) |
| 29 | \( 1 - 104128242846 T + p^{15} T^{2} \) |
| 31 | \( 1 - 296696681512 T + p^{15} T^{2} \) |
| 37 | \( 1 + 178337455666 T + p^{15} T^{2} \) |
| 41 | \( 1 + 1790882416086 T + p^{15} T^{2} \) |
| 43 | \( 1 - 2863459422772 T + p^{15} T^{2} \) |
| 47 | \( 1 + 4332907521600 T + p^{15} T^{2} \) |
| 53 | \( 1 - 9732317104422 T + p^{15} T^{2} \) |
| 59 | \( 1 - 13514837176500 T + p^{15} T^{2} \) |
| 61 | \( 1 - 5352663511190 T + p^{15} T^{2} \) |
| 67 | \( 1 - 53233909720108 T + p^{15} T^{2} \) |
| 71 | \( 1 - 20229661643400 T + p^{15} T^{2} \) |
| 73 | \( 1 - 26264166466106 T + p^{15} T^{2} \) |
| 79 | \( 1 - 339031361615128 T + p^{15} T^{2} \) |
| 83 | \( 1 + 131684771045076 T + p^{15} T^{2} \) |
| 89 | \( 1 + 39352148322678 T + p^{15} T^{2} \) |
| 97 | \( 1 - 1128750908801474 T + p^{15} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.18413160531820470443636987333, −11.54278934596006628672653699170, −10.09419220568613983750186054585, −8.626899562696338881482010659309, −7.936155264521876388757542235271, −6.56532092414085753268639854631, −4.69634048940393689187259380409, −3.81961601651677936341923737479, −2.12980675858836377175225614020, −0.963813282095726779369554596289,
0.963813282095726779369554596289, 2.12980675858836377175225614020, 3.81961601651677936341923737479, 4.69634048940393689187259380409, 6.56532092414085753268639854631, 7.936155264521876388757542235271, 8.626899562696338881482010659309, 10.09419220568613983750186054585, 11.54278934596006628672653699170, 12.18413160531820470443636987333
