| L(s) = 1 | + 2.43·2-s − 0.753·3-s + 3.91·4-s − 0.341·5-s − 1.83·6-s + 4.65·8-s − 2.43·9-s − 0.830·10-s − 2.43·11-s − 2.95·12-s + 0.257·15-s + 3.49·16-s + 1.94·17-s − 5.91·18-s + 6.29·19-s − 1.33·20-s − 5.91·22-s − 3.68·23-s − 3.51·24-s − 4.88·25-s + 4.09·27-s + 4.44·29-s + 0.625·30-s + 1.97·31-s − 0.808·32-s + 1.83·33-s + 4.73·34-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 0.435·3-s + 1.95·4-s − 0.152·5-s − 0.748·6-s + 1.64·8-s − 0.810·9-s − 0.262·10-s − 0.733·11-s − 0.851·12-s + 0.0664·15-s + 0.874·16-s + 0.472·17-s − 1.39·18-s + 1.44·19-s − 0.298·20-s − 1.26·22-s − 0.769·23-s − 0.716·24-s − 0.976·25-s + 0.787·27-s + 0.824·29-s + 0.114·30-s + 0.354·31-s − 0.142·32-s + 0.319·33-s + 0.812·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 3 | \( 1 + 0.753T + 3T^{2} \) |
| 5 | \( 1 + 0.341T + 5T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + 9.62T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 8.40T + 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 + 0.626T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 5.59T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.11575451870875000498864428105, −6.58789194792514989070804421125, −5.63182047636269942871347210789, −5.46314690812460276598635946662, −4.83486699147128412982876641762, −3.89820182628633810397672851875, −3.22275343116121289590932936374, −2.66272271148759598725959920376, −1.58149951382022750043497983530, 0,
1.58149951382022750043497983530, 2.66272271148759598725959920376, 3.22275343116121289590932936374, 3.89820182628633810397672851875, 4.83486699147128412982876641762, 5.46314690812460276598635946662, 5.63182047636269942871347210789, 6.58789194792514989070804421125, 7.11575451870875000498864428105
