| L(s) = 1 | + 2.95·3-s − 2.20·5-s + 2.79·7-s + 5.72·9-s − 1.59·11-s − 3.09·13-s − 6.52·15-s − 6.40·17-s − 3.68·19-s + 8.24·21-s + 3.82·23-s − 0.117·25-s + 8.04·27-s + 1.84·29-s − 9.59·31-s − 4.70·33-s − 6.16·35-s − 9.44·37-s − 9.14·39-s − 9.33·41-s − 2.49·43-s − 12.6·45-s − 2.81·47-s + 0.789·49-s − 18.9·51-s + 1.77·53-s + 3.51·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s − 0.988·5-s + 1.05·7-s + 1.90·9-s − 0.480·11-s − 0.858·13-s − 1.68·15-s − 1.55·17-s − 0.846·19-s + 1.79·21-s + 0.798·23-s − 0.0235·25-s + 1.54·27-s + 0.343·29-s − 1.72·31-s − 0.818·33-s − 1.04·35-s − 1.55·37-s − 1.46·39-s − 1.45·41-s − 0.381·43-s − 1.88·45-s − 0.410·47-s + 0.112·49-s − 2.64·51-s + 0.243·53-s + 0.474·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.95T + 3T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 - 1.84T + 29T^{2} \) |
| 31 | \( 1 + 9.59T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 + 9.33T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 + 2.81T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 - 7.92T + 67T^{2} \) |
| 71 | \( 1 - 0.0832T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 9.44T + 97T^{2} \) |
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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
−7.905523911453740412004446724334, −7.17847682356080410205053891666, −6.82983539122461240954525812087, −5.20656750305105644380686828401, −4.65491234204823456611339399858, −3.92721907136378165571141873405, −3.26298111748844213434357452544, −2.24761997036934579706047577147, −1.81293191334837924359326119250, 0,
1.81293191334837924359326119250, 2.24761997036934579706047577147, 3.26298111748844213434357452544, 3.92721907136378165571141873405, 4.65491234204823456611339399858, 5.20656750305105644380686828401, 6.82983539122461240954525812087, 7.17847682356080410205053891666, 7.905523911453740412004446724334
