L(s) = 1 | + 2-s + 0.537·3-s + 4-s + 5-s + 0.537·6-s − 0.123·7-s + 8-s − 2.71·9-s + 10-s − 1.46·11-s + 0.537·12-s − 1.53·13-s − 0.123·14-s + 0.537·15-s + 16-s + 0.711·17-s − 2.71·18-s − 8.20·19-s + 20-s − 0.0661·21-s − 1.46·22-s + 0.537·24-s + 25-s − 1.53·26-s − 3.06·27-s − 0.123·28-s + 2.17·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.310·3-s + 0.5·4-s + 0.447·5-s + 0.219·6-s − 0.0465·7-s + 0.353·8-s − 0.903·9-s + 0.316·10-s − 0.441·11-s + 0.155·12-s − 0.426·13-s − 0.0328·14-s + 0.138·15-s + 0.250·16-s + 0.172·17-s − 0.639·18-s − 1.88·19-s + 0.223·20-s − 0.0144·21-s − 0.311·22-s + 0.109·24-s + 0.200·25-s − 0.301·26-s − 0.590·27-s − 0.0232·28-s + 0.403·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 0.537T + 3T^{2} \) |
| 7 | \( 1 + 0.123T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 0.711T + 17T^{2} \) |
| 19 | \( 1 + 8.20T + 19T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 + 0.562T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 + 6.86T + 67T^{2} \) |
| 71 | \( 1 + 2.46T + 71T^{2} \) |
| 73 | \( 1 + 9.49T + 73T^{2} \) |
| 79 | \( 1 - 6.35T + 79T^{2} \) |
| 83 | \( 1 + 2.17T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.931055069640215612622755370804, −6.94387453305249287311061614183, −6.26836982273769984873775954151, −5.68485582692229406780677286043, −4.89037218428278846624126176156, −4.18800760607272509917630891287, −3.15024776019384694415772400033, −2.54991332127767910706772773438, −1.73387243808139653254933456719, 0,
1.73387243808139653254933456719, 2.54991332127767910706772773438, 3.15024776019384694415772400033, 4.18800760607272509917630891287, 4.89037218428278846624126176156, 5.68485582692229406780677286043, 6.26836982273769984873775954151, 6.94387453305249287311061614183, 7.931055069640215612622755370804