| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s − 11-s − 4·13-s + 2·14-s + 16-s − 3.46·17-s + 7.19·19-s − 20-s − 22-s − 5.19·23-s + 25-s − 4·26-s + 2·28-s − 9·29-s + 8.46·31-s + 32-s − 3.46·34-s − 2·35-s + 2.92·37-s + 7.19·38-s − 40-s − 2.53·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 1.10·13-s + 0.534·14-s + 0.250·16-s − 0.840·17-s + 1.65·19-s − 0.223·20-s − 0.213·22-s − 1.08·23-s + 0.200·25-s − 0.784·26-s + 0.377·28-s − 1.67·29-s + 1.52·31-s + 0.176·32-s − 0.594·34-s − 0.338·35-s + 0.481·37-s + 1.16·38-s − 0.158·40-s − 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 8.46T + 31T^{2} \) |
| 37 | \( 1 - 2.92T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 5.73T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 2.26T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 9.92T + 83T^{2} \) |
| 89 | \( 1 + 6.80T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.31234108330173802767184683117, −6.83894783392441638571416815219, −5.84877773048307861784376338494, −5.17772113190884265253531289831, −4.68282559809824548592304654713, −3.97009155906140707174843490242, −3.10411887967950057761334040170, −2.32964482916759901977993028102, −1.44817818537124017814808476810, 0,
1.44817818537124017814808476810, 2.32964482916759901977993028102, 3.10411887967950057761334040170, 3.97009155906140707174843490242, 4.68282559809824548592304654713, 5.17772113190884265253531289831, 5.84877773048307861784376338494, 6.83894783392441638571416815219, 7.31234108330173802767184683117
