| L(s) = 1 | − 0.229·2-s + 0.955·3-s − 0.947·4-s + 1.21·5-s − 0.219·6-s + 0.380·7-s + 0.446·8-s − 0.0871·9-s − 0.278·10-s + 0.0766·11-s − 0.905·12-s − 0.0873·14-s + 1.15·15-s + 0.844·16-s + 0.0199·18-s − 1.14·20-s + 0.363·21-s − 0.0175·22-s + 0.426·24-s + 0.470·25-s − 1.03·27-s − 0.360·28-s + 1.44·29-s − 0.265·30-s − 0.640·32-s + 0.0731·33-s + 0.461·35-s + ⋯ |
| L(s) = 1 | − 0.229·2-s + 0.955·3-s − 0.947·4-s + 1.21·5-s − 0.219·6-s + 0.380·7-s + 0.446·8-s − 0.0871·9-s − 0.278·10-s + 0.0766·11-s − 0.905·12-s − 0.0873·14-s + 1.15·15-s + 0.844·16-s + 0.0199·18-s − 1.14·20-s + 0.363·21-s − 0.0175·22-s + 0.426·24-s + 0.470·25-s − 1.03·27-s − 0.360·28-s + 1.44·29-s − 0.265·30-s − 0.640·32-s + 0.0731·33-s + 0.461·35-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.275641739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.275641739\) |
| \(L(1)\) |
|
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 | \( 1 + 0.229T + T^{2} \) |
| 3 | \( 1 - 0.955T + T^{2} \) |
| 5 | \( 1 - 1.21T + T^{2} \) |
| 7 | \( 1 - 0.380T + T^{2} \) |
| 11 | \( 1 - 0.0766T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.44T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.33T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.676T + T^{2} \) |
| 47 | \( 1 + 1.71T + T^{2} \) |
| 53 | \( 1 - 1.63T + T^{2} \) |
| 59 | \( 1 + 0.529T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.90T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.85T + T^{2} \) |
| 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
−9.839684114730098601276475862404, −9.114993614354630761482503528446, −8.518223462696827659532033763420, −7.915979697775129061531757848643, −6.70180823037142418891302335425, −5.63195505203823723940266107018, −4.90008896855319099893278572182, −3.75938727296407984820069598728, −2.65888237942521538726245787027, −1.54703238992230805996035204824,
1.54703238992230805996035204824, 2.65888237942521538726245787027, 3.75938727296407984820069598728, 4.90008896855319099893278572182, 5.63195505203823723940266107018, 6.70180823037142418891302335425, 7.915979697775129061531757848643, 8.518223462696827659532033763420, 9.114993614354630761482503528446, 9.839684114730098601276475862404
