| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 6·11-s + 13-s + 14-s + 16-s + 8·17-s + 20-s − 6·22-s + 25-s − 26-s − 28-s − 6·29-s − 2·31-s − 32-s − 8·34-s − 35-s + 10·37-s − 40-s + 8·41-s − 6·43-s + 6·44-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.359·31-s − 0.176·32-s − 1.37·34-s − 0.169·35-s + 1.64·37-s − 0.158·40-s + 1.24·41-s − 0.914·43-s + 0.904·44-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.042894760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042894760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.81551071327692358549223132112, −7.16995971415067238005494469368, −6.46799980397104600387712348320, −5.91723260977663932706436973688, −5.27567478175236568488675063070, −3.97555935495980417951154800106, −3.55835990693938135065972997346, −2.53599393485021238061585608000, −1.48700557028494144262365127160, −0.876602218626080538967550499322,
0.876602218626080538967550499322, 1.48700557028494144262365127160, 2.53599393485021238061585608000, 3.55835990693938135065972997346, 3.97555935495980417951154800106, 5.27567478175236568488675063070, 5.91723260977663932706436973688, 6.46799980397104600387712348320, 7.16995971415067238005494469368, 7.81551071327692358549223132112
