| L(s) = 1 | − 176.·2-s + 573.·3-s + 2.28e4·4-s − 1.01e5·6-s + 2.01e5·7-s − 2.58e6·8-s − 1.26e6·9-s − 3.34e6·11-s + 1.31e7·12-s + 7.80e6·13-s − 3.55e7·14-s + 2.68e8·16-s + 8.71e7·17-s + 2.23e8·18-s − 1.66e7·19-s + 1.15e8·21-s + 5.89e8·22-s − 1.13e9·23-s − 1.48e9·24-s − 1.37e9·26-s − 1.63e9·27-s + 4.60e9·28-s + 2.60e9·29-s + 8.33e8·31-s − 2.60e10·32-s − 1.91e9·33-s − 1.53e10·34-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.453·3-s + 2.79·4-s − 0.883·6-s + 0.647·7-s − 3.48·8-s − 0.793·9-s − 0.569·11-s + 1.26·12-s + 0.448·13-s − 1.26·14-s + 3.99·16-s + 0.875·17-s + 1.54·18-s − 0.0813·19-s + 0.293·21-s + 1.10·22-s − 1.59·23-s − 1.58·24-s − 0.873·26-s − 0.814·27-s + 1.80·28-s + 0.813·29-s + 0.168·31-s − 4.29·32-s − 0.258·33-s − 1.70·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 176.T + 8.19e3T^{2} \) |
| 3 | \( 1 - 573.T + 1.59e6T^{2} \) |
| 7 | \( 1 - 2.01e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 3.34e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 7.80e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 8.71e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.66e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.13e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.60e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 8.33e8T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.05e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.33e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.93e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 2.85e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.23e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 5.55e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 4.10e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 3.36e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.57e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.05e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 6.93e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.01e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 8.51e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 7.99e12T + 6.73e25T^{2} \) |
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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
−14.29015555997247277226935340099, −11.99327076977578146120722372151, −10.89910403216659843849711366816, −9.728256485772880488676437550835, −8.388791523467682243214165664089, −7.80959815298439845351608659115, −6.02160125398896531640036726094, −2.88404933917070394186278226849, −1.55042875338057565409250737663, 0,
1.55042875338057565409250737663, 2.88404933917070394186278226849, 6.02160125398896531640036726094, 7.80959815298439845351608659115, 8.388791523467682243214165664089, 9.728256485772880488676437550835, 10.89910403216659843849711366816, 11.99327076977578146120722372151, 14.29015555997247277226935340099
