| L(s) = 1 | − 22.4·2-s + 2.28e3·3-s − 7.68e3·4-s + 1.79e4·5-s − 5.12e4·6-s − 5.63e5·7-s + 3.56e5·8-s + 3.62e6·9-s − 4.04e5·10-s + 6.98e6·11-s − 1.75e7·12-s + 1.26e7·14-s + 4.11e7·15-s + 5.49e7·16-s − 3.79e7·17-s − 8.13e7·18-s + 6.56e7·19-s − 1.38e8·20-s − 1.28e9·21-s − 1.56e8·22-s − 9.97e8·23-s + 8.14e8·24-s − 8.96e8·25-s + 4.63e9·27-s + 4.32e9·28-s − 3.48e9·29-s − 9.23e8·30-s + ⋯ |
| L(s) = 1 | − 0.248·2-s + 1.80·3-s − 0.938·4-s + 0.515·5-s − 0.448·6-s − 1.80·7-s + 0.480·8-s + 2.27·9-s − 0.127·10-s + 1.18·11-s − 1.69·12-s + 0.448·14-s + 0.931·15-s + 0.819·16-s − 0.381·17-s − 0.563·18-s + 0.319·19-s − 0.483·20-s − 3.27·21-s − 0.295·22-s − 1.40·23-s + 0.870·24-s − 0.734·25-s + 2.30·27-s + 1.69·28-s − 1.08·29-s − 0.231·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 + 22.4T + 8.19e3T^{2} \) |
| 3 | \( 1 - 2.28e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 1.79e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.63e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 6.98e6T + 3.45e13T^{2} \) |
| 17 | \( 1 + 3.79e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 6.56e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 9.97e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.48e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 2.61e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 8.62e8T + 2.43e20T^{2} \) |
| 41 | \( 1 - 1.77e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 6.99e8T + 1.71e21T^{2} \) |
| 47 | \( 1 - 4.91e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.91e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 2.32e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.05e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 8.74e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 9.60e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 8.68e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 7.23e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.62e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 4.90e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.50e12T + 6.73e25T^{2} \) |
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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
−9.503194214598205949127523119644, −9.230519903150653897755582027013, −8.214029220323887153864121221920, −7.08385390640742772300195336913, −5.97467429966832697641300203628, −4.05351502932441792682647219271, −3.65305973344435308581704633054, −2.51029801477733607865894527372, −1.38091367591751823327998194371, 0,
1.38091367591751823327998194371, 2.51029801477733607865894527372, 3.65305973344435308581704633054, 4.05351502932441792682647219271, 5.97467429966832697641300203628, 7.08385390640742772300195336913, 8.214029220323887153864121221920, 9.230519903150653897755582027013, 9.503194214598205949127523119644
