L(s) = 1 | − 1.54·2-s + 1.97·3-s + 1.38·4-s + 1.44·5-s − 3.04·6-s − 0.818·7-s − 0.587·8-s + 2.90·9-s − 2.22·10-s − 1.94·11-s + 2.72·12-s + 1.26·14-s + 2.84·15-s − 0.474·16-s − 4.48·18-s + 1.98·20-s − 1.61·21-s + 3.00·22-s − 1.16·24-s + 1.07·25-s + 3.76·27-s − 1.12·28-s − 0.229·29-s − 4.39·30-s + 1.31·32-s − 3.84·33-s − 1.17·35-s + ⋯ |
L(s) = 1 | − 1.54·2-s + 1.97·3-s + 1.38·4-s + 1.44·5-s − 3.04·6-s − 0.818·7-s − 0.587·8-s + 2.90·9-s − 2.22·10-s − 1.94·11-s + 2.72·12-s + 1.26·14-s + 2.84·15-s − 0.474·16-s − 4.48·18-s + 1.98·20-s − 1.61·21-s + 3.00·22-s − 1.16·24-s + 1.07·25-s + 3.76·27-s − 1.12·28-s − 0.229·29-s − 4.39·30-s + 1.31·32-s − 3.84·33-s − 1.17·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.078505679\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078505679\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1151 | \( 1 - T \) |
good | 2 | \( 1 + 1.54T + T^{2} \) |
| 3 | \( 1 - 1.97T + T^{2} \) |
| 5 | \( 1 - 1.44T + T^{2} \) |
| 7 | \( 1 + 0.818T + T^{2} \) |
| 11 | \( 1 + 1.94T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.229T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.676T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.955T + T^{2} \) |
| 47 | \( 1 + 1.99T + T^{2} \) |
| 53 | \( 1 + 1.71T + T^{2} \) |
| 59 | \( 1 - 0.0766T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.529T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.33T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.747326391434078590997297766536, −9.334206097216429184821632174820, −8.464325720629819381405963240934, −7.88446790609264200966899479631, −7.17789708275969238427277351754, −6.19796338862628704395842553067, −4.77974097110852192866667936544, −3.10102448780345108918181793998, −2.48599596464021658242184605788, −1.67491823708135153088302467569,
1.67491823708135153088302467569, 2.48599596464021658242184605788, 3.10102448780345108918181793998, 4.77974097110852192866667936544, 6.19796338862628704395842553067, 7.17789708275969238427277351754, 7.88446790609264200966899479631, 8.464325720629819381405963240934, 9.334206097216429184821632174820, 9.747326391434078590997297766536