| L(s) = 1 | − 3-s − 4.62·7-s + 9-s − 4.94·11-s + 3.76·13-s + 2.69·17-s + 5.87·19-s + 4.62·21-s − 6.67·23-s − 27-s + 1.20·29-s + 3.30·31-s + 4.94·33-s − 1.87·37-s − 3.76·39-s − 3.03·41-s + 10.6·43-s − 0.259·47-s + 14.4·49-s − 2.69·51-s − 9.79·53-s − 5.87·57-s − 9.62·59-s + 6.27·61-s − 4.62·63-s − 2.56·67-s + 6.67·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.74·7-s + 0.333·9-s − 1.49·11-s + 1.04·13-s + 0.652·17-s + 1.34·19-s + 1.00·21-s − 1.39·23-s − 0.192·27-s + 0.222·29-s + 0.593·31-s + 0.861·33-s − 0.308·37-s − 0.602·39-s − 0.473·41-s + 1.62·43-s − 0.0378·47-s + 2.05·49-s − 0.376·51-s − 1.34·53-s − 0.777·57-s − 1.25·59-s + 0.803·61-s − 0.582·63-s − 0.312·67-s + 0.803·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 - 1.20T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 0.259T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 + 9.62T + 59T^{2} \) |
| 61 | \( 1 - 6.27T + 61T^{2} \) |
| 67 | \( 1 + 2.56T + 67T^{2} \) |
| 71 | \( 1 - 8.67T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 8.89T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 3.98T + 97T^{2} \) |
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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
−7.65749133761984581522710840976, −6.65242112264308161591655125957, −6.10604279724955999744592969861, −5.62437970448515691793624662977, −4.85740315324804958935006437063, −3.72400856893289305488004543788, −3.25384611242444307568401727463, −2.39760497070489063614347563917, −0.996494078002450650416353986526, 0,
0.996494078002450650416353986526, 2.39760497070489063614347563917, 3.25384611242444307568401727463, 3.72400856893289305488004543788, 4.85740315324804958935006437063, 5.62437970448515691793624662977, 6.10604279724955999744592969861, 6.65242112264308161591655125957, 7.65749133761984581522710840976
