L(s) = 1 | + 0.525·2-s − 3-s − 1.72·4-s − 0.525·6-s + 7-s − 1.95·8-s + 9-s − 11-s + 1.72·12-s + 4.97·13-s + 0.525·14-s + 2.41·16-s − 3.11·17-s + 0.525·18-s − 7.37·19-s − 21-s − 0.525·22-s + 0.0645·23-s + 1.95·24-s + 2.61·26-s − 27-s − 1.72·28-s + 1.75·29-s + 9.15·31-s + 5.18·32-s + 33-s − 1.63·34-s + ⋯ |
L(s) = 1 | + 0.371·2-s − 0.577·3-s − 0.861·4-s − 0.214·6-s + 0.377·7-s − 0.692·8-s + 0.333·9-s − 0.301·11-s + 0.497·12-s + 1.37·13-s + 0.140·14-s + 0.604·16-s − 0.754·17-s + 0.123·18-s − 1.69·19-s − 0.218·21-s − 0.112·22-s + 0.0134·23-s + 0.399·24-s + 0.512·26-s − 0.192·27-s − 0.325·28-s + 0.325·29-s + 1.64·31-s + 0.916·32-s + 0.174·33-s − 0.280·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 0.525T + 2T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 + 3.11T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 - 0.0645T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 - 9.15T + 31T^{2} \) |
| 37 | \( 1 + 7.77T + 37T^{2} \) |
| 41 | \( 1 + 9.75T + 41T^{2} \) |
| 43 | \( 1 - 1.51T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 2.59T + 73T^{2} \) |
| 79 | \( 1 - 2.00T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 + 5.46T + 97T^{2} \) |
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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
−7.941136814572491399308429353572, −6.72408582195197465733967814322, −6.30253182018302117961901506280, −5.53173998812369639042394689446, −4.78585615164671357401802052985, −4.22414145920617143380345097624, −3.53691376181256640305141568927, −2.34992638549978341743329715396, −1.17307473951648607728130108912, 0,
1.17307473951648607728130108912, 2.34992638549978341743329715396, 3.53691376181256640305141568927, 4.22414145920617143380345097624, 4.78585615164671357401802052985, 5.53173998812369639042394689446, 6.30253182018302117961901506280, 6.72408582195197465733967814322, 7.941136814572491399308429353572