| L(s) = 1 | + 13.2·2-s + 17.5·3-s + 46.3·4-s + 17.9·5-s + 232.·6-s − 343·7-s − 1.07e3·8-s − 1.87e3·9-s + 236.·10-s + 3.12e3·11-s + 815.·12-s + 1.42e4·13-s − 4.52e3·14-s + 315.·15-s − 2.01e4·16-s − 5.50e3·17-s − 2.47e4·18-s + 2.90e4·19-s + 832.·20-s − 6.03e3·21-s + 4.13e4·22-s − 3.99e3·23-s − 1.89e4·24-s − 7.78e4·25-s + 1.88e5·26-s − 7.14e4·27-s − 1.59e4·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.376·3-s + 0.362·4-s + 0.0642·5-s + 0.439·6-s − 0.377·7-s − 0.744·8-s − 0.858·9-s + 0.0749·10-s + 0.708·11-s + 0.136·12-s + 1.79·13-s − 0.441·14-s + 0.0241·15-s − 1.23·16-s − 0.271·17-s − 1.00·18-s + 0.970·19-s + 0.0232·20-s − 0.142·21-s + 0.827·22-s − 0.0684·23-s − 0.279·24-s − 0.995·25-s + 2.09·26-s − 0.699·27-s − 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.104788213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.104788213\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| good | 2 | \( 1 - 13.2T + 128T^{2} \) |
| 3 | \( 1 - 17.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 17.9T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.12e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.42e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.50e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.90e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.99e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.32e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.15e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.50e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.14e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.13e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.71e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.27e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.54e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.77e6T + 8.07e13T^{2} \) |
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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
−21.14357199722605128099117789009, −19.85703242294188046364948049321, −17.99615491591178053259269306805, −15.92886992242629341607193624881, −14.33278475838902338493712788027, −13.35729929638836271104508273974, −11.59040495312761125574552358148, −8.930689947234920834672850754946, −5.98912370532902750246157725918, −3.54736914293094747875810162021,
3.54736914293094747875810162021, 5.98912370532902750246157725918, 8.930689947234920834672850754946, 11.59040495312761125574552358148, 13.35729929638836271104508273974, 14.33278475838902338493712788027, 15.92886992242629341607193624881, 17.99615491591178053259269306805, 19.85703242294188046364948049321, 21.14357199722605128099117789009
