| L(s) = 1 | + 3-s − 4.06·5-s + 1.52·7-s + 9-s + 3.75·11-s − 0.808·13-s − 4.06·15-s − 1.48·17-s + 5.96·19-s + 1.52·21-s + 8.26·23-s + 11.4·25-s + 27-s − 5.45·29-s − 8.18·31-s + 3.75·33-s − 6.18·35-s − 5.32·37-s − 0.808·39-s + 5.14·41-s + 5.00·43-s − 4.06·45-s − 3.25·47-s − 4.68·49-s − 1.48·51-s + 7.56·53-s − 15.2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.81·5-s + 0.575·7-s + 0.333·9-s + 1.13·11-s − 0.224·13-s − 1.04·15-s − 0.359·17-s + 1.36·19-s + 0.332·21-s + 1.72·23-s + 2.29·25-s + 0.192·27-s − 1.01·29-s − 1.46·31-s + 0.652·33-s − 1.04·35-s − 0.875·37-s − 0.129·39-s + 0.803·41-s + 0.763·43-s − 0.605·45-s − 0.474·47-s − 0.669·49-s − 0.207·51-s + 1.03·53-s − 2.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.050129612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.050129612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 4.06T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 + 0.808T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 - 8.26T + 23T^{2} \) |
| 29 | \( 1 + 5.45T + 29T^{2} \) |
| 31 | \( 1 + 8.18T + 31T^{2} \) |
| 37 | \( 1 + 5.32T + 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 43 | \( 1 - 5.00T + 43T^{2} \) |
| 47 | \( 1 + 3.25T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 0.362T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 2.55T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 6.54T + 97T^{2} \) |
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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
−7.63570845450164399688941725157, −7.29940151769250657471896065314, −6.91865550695439402025456717140, −5.60352120451546106067402229563, −4.82585798885069369088496501291, −4.14278755335285176820879134035, −3.55690368594374562062022891383, −2.96758059348997047020023986949, −1.66160280222778049600515639783, −0.72258508544076634876454735121,
0.72258508544076634876454735121, 1.66160280222778049600515639783, 2.96758059348997047020023986949, 3.55690368594374562062022891383, 4.14278755335285176820879134035, 4.82585798885069369088496501291, 5.60352120451546106067402229563, 6.91865550695439402025456717140, 7.29940151769250657471896065314, 7.63570845450164399688941725157
