| L(s) = 1 | + 4.09e3·2-s − 1.16e5·3-s + 1.67e7·4-s − 4.77e8·6-s + 1.34e10·7-s + 6.87e10·8-s − 8.33e11·9-s + 4.67e11·11-s − 1.95e12·12-s + 9.38e13·13-s + 5.50e13·14-s + 2.81e14·16-s − 6.14e14·17-s − 3.41e15·18-s − 1.66e16·19-s − 1.56e15·21-s + 1.91e15·22-s + 1.60e17·23-s − 8.01e15·24-s + 3.84e17·26-s + 1.96e17·27-s + 2.25e17·28-s − 1.05e18·29-s − 3.00e18·31-s + 1.15e18·32-s − 5.45e16·33-s − 2.51e18·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.126·3-s + 0.5·4-s − 0.0896·6-s + 0.366·7-s + 0.353·8-s − 0.983·9-s + 0.0449·11-s − 0.0633·12-s + 1.11·13-s + 0.259·14-s + 0.250·16-s − 0.255·17-s − 0.695·18-s − 1.72·19-s − 0.0465·21-s + 0.0317·22-s + 1.53·23-s − 0.0448·24-s + 0.790·26-s + 0.251·27-s + 0.183·28-s − 0.551·29-s − 0.685·31-s + 0.176·32-s − 0.00569·33-s − 0.180·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.16e5T + 8.47e11T^{2} \) |
| 7 | \( 1 - 1.34e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 4.67e11T + 1.08e26T^{2} \) |
| 13 | \( 1 - 9.38e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 6.14e14T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.66e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.60e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.05e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.00e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 3.36e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 5.09e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 2.92e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 9.77e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 5.26e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 4.10e20T + 1.86e44T^{2} \) |
| 61 | \( 1 + 5.44e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 6.32e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.81e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.48e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 3.43e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 6.85e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 4.29e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 2.90e24T + 4.66e49T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.90016275892947585620938112617, −9.054234944149070459552504625502, −8.133753180465471118332240056815, −6.67523736490402107349380770004, −5.81878266763209471720874419260, −4.73131741716939731113093320322, −3.62227231481011207344262671958, −2.52453471070194034842298491789, −1.36542102772041476671398671123, 0,
1.36542102772041476671398671123, 2.52453471070194034842298491789, 3.62227231481011207344262671958, 4.73131741716939731113093320322, 5.81878266763209471720874419260, 6.67523736490402107349380770004, 8.133753180465471118332240056815, 9.054234944149070459552504625502, 10.90016275892947585620938112617
