| L(s) = 1 | − 1.68·3-s − 3.32·5-s + 3.49·7-s − 0.173·9-s + 1.26·11-s − 1.98·13-s + 5.58·15-s − 1.03·17-s + 19-s − 5.87·21-s − 6.45·23-s + 6.02·25-s + 5.33·27-s + 5.24·29-s − 0.427·31-s − 2.12·33-s − 11.6·35-s + 8.20·37-s + 3.33·39-s − 12.4·41-s − 4.70·43-s + 0.575·45-s + 9.04·47-s + 5.22·49-s + 1.73·51-s + 0.909·53-s − 4.20·55-s + ⋯ |
| L(s) = 1 | − 0.970·3-s − 1.48·5-s + 1.32·7-s − 0.0578·9-s + 0.381·11-s − 0.550·13-s + 1.44·15-s − 0.250·17-s + 0.229·19-s − 1.28·21-s − 1.34·23-s + 1.20·25-s + 1.02·27-s + 0.973·29-s − 0.0767·31-s − 0.370·33-s − 1.96·35-s + 1.34·37-s + 0.533·39-s − 1.94·41-s − 0.718·43-s + 0.0858·45-s + 1.31·47-s + 0.746·49-s + 0.243·51-s + 0.124·53-s − 0.567·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 1.98T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 + 0.427T + 31T^{2} \) |
| 37 | \( 1 - 8.20T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 - 0.909T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 3.76T + 67T^{2} \) |
| 71 | \( 1 - 5.33T + 71T^{2} \) |
| 73 | \( 1 - 5.75T + 73T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.82983088752623947215568720642, −7.05400282950216529393786307227, −6.33748927272201247138596384912, −5.44284788719919250006058799980, −4.74668885854992284571992451840, −4.31011357489466134542689210043, −3.42265691973845571234475851177, −2.24389218972198047985699598431, −1.02664138539665128640691545062, 0,
1.02664138539665128640691545062, 2.24389218972198047985699598431, 3.42265691973845571234475851177, 4.31011357489466134542689210043, 4.74668885854992284571992451840, 5.44284788719919250006058799980, 6.33748927272201247138596384912, 7.05400282950216529393786307227, 7.82983088752623947215568720642
