| L(s) = 1 | + 23.4·2-s + 729·3-s − 7.64e3·4-s − 4.29e4·5-s + 1.71e4·6-s + 4.65e5·7-s − 3.71e5·8-s + 5.31e5·9-s − 1.00e6·10-s − 5.09e6·11-s − 5.57e6·12-s − 1.16e7·13-s + 1.09e7·14-s − 3.13e7·15-s + 5.38e7·16-s − 1.31e8·17-s + 1.24e7·18-s + 3.44e8·19-s + 3.28e8·20-s + 3.39e8·21-s − 1.19e8·22-s + 8.75e8·23-s − 2.70e8·24-s + 6.23e8·25-s − 2.72e8·26-s + 3.87e8·27-s − 3.55e9·28-s + ⋯ |
| L(s) = 1 | + 0.259·2-s + 0.577·3-s − 0.932·4-s − 1.22·5-s + 0.149·6-s + 1.49·7-s − 0.501·8-s + 0.333·9-s − 0.318·10-s − 0.866·11-s − 0.538·12-s − 0.667·13-s + 0.387·14-s − 0.709·15-s + 0.802·16-s − 1.32·17-s + 0.0864·18-s + 1.67·19-s + 1.14·20-s + 0.863·21-s − 0.224·22-s + 1.23·23-s − 0.289·24-s + 0.510·25-s − 0.173·26-s + 0.192·27-s − 1.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 729T \) |
| 59 | \( 1 - 4.21e10T \) |
| good | 2 | \( 1 - 23.4T + 8.19e3T^{2} \) |
| 5 | \( 1 + 4.29e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 4.65e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.09e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.16e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.31e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.44e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 8.75e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 1.80e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 4.43e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.28e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.03e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.91e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.00e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 9.81e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 6.23e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 9.29e10T + 5.48e23T^{2} \) |
| 71 | \( 1 - 8.86e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 5.94e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.79e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 6.61e10T + 8.87e24T^{2} \) |
| 89 | \( 1 + 2.43e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 8.12e12T + 6.73e25T^{2} \) |
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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
−9.666611739697829701599102838126, −8.604204853151723595820297657501, −7.965287258461326992887720847642, −7.24642205704936311890678733610, −5.12991963277705198100096940101, −4.73327721877430648141555179597, −3.69357605320598498819178213907, −2.58376338772635386903941279732, −1.09596995878708173717524687149, 0,
1.09596995878708173717524687149, 2.58376338772635386903941279732, 3.69357605320598498819178213907, 4.73327721877430648141555179597, 5.12991963277705198100096940101, 7.24642205704936311890678733610, 7.965287258461326992887720847642, 8.604204853151723595820297657501, 9.666611739697829701599102838126
