L(s) = 1 | + (−0.580 + 0.814i)2-s + (0.550 + 0.353i)3-s + (−0.327 − 0.945i)4-s + (−0.142 − 0.989i)5-s + (−0.607 + 0.243i)6-s + (0.0947 + 0.00904i)7-s + (0.959 + 0.281i)8-s + (−0.237 − 0.520i)9-s + (0.888 + 0.458i)10-s + (0.154 − 0.635i)12-s + (−0.0623 + 0.0719i)14-s + (0.271 − 0.595i)15-s + (−0.786 + 0.618i)16-s + (0.561 + 0.108i)18-s + (−0.888 + 0.458i)20-s + (0.0489 + 0.0384i)21-s + ⋯ |
L(s) = 1 | + (−0.580 + 0.814i)2-s + (0.550 + 0.353i)3-s + (−0.327 − 0.945i)4-s + (−0.142 − 0.989i)5-s + (−0.607 + 0.243i)6-s + (0.0947 + 0.00904i)7-s + (0.959 + 0.281i)8-s + (−0.237 − 0.520i)9-s + (0.888 + 0.458i)10-s + (0.154 − 0.635i)12-s + (−0.0623 + 0.0719i)14-s + (0.271 − 0.595i)15-s + (−0.786 + 0.618i)16-s + (0.561 + 0.108i)18-s + (−0.888 + 0.458i)20-s + (0.0489 + 0.0384i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8740251701\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8740251701\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.580 - 0.814i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (0.981 + 0.189i)T \) |
good | 3 | \( 1 + (-0.550 - 0.353i)T + (0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.0947 - 0.00904i)T + (0.981 + 0.189i)T^{2} \) |
| 11 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 13 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 17 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 19 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 23 | \( 1 + (-0.0475 + 0.998i)T + (-0.995 - 0.0950i)T^{2} \) |
| 29 | \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.42 + 0.273i)T + (0.928 - 0.371i)T^{2} \) |
| 43 | \( 1 + (-1.28 - 1.48i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.419 + 0.216i)T + (0.580 - 0.814i)T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 61 | \( 1 + (0.264 - 0.105i)T + (0.723 - 0.690i)T^{2} \) |
| 71 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 73 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 79 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 83 | \( 1 + (1.56 - 1.23i)T + (0.235 - 0.971i)T^{2} \) |
| 89 | \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.458423691385647256716470101817, −8.966073673074550023028265348524, −8.216684423892695910059583994208, −7.70128656546207715891286722548, −6.43939631612172022530869172372, −5.81058586754022696530742930499, −4.65762075179875983803483235401, −4.10127233613257211463284012236, −2.51311552361710309808480196884, −0.900720334353485672056941410789,
1.63859914280516492495856599076, 2.68650783084321810334333765051, 3.30517133692585945645084083635, 4.42926296552179552158155345677, 5.70379047259414540488412021379, 7.05860236833744169372215958986, 7.47440939156441138011232552886, 8.295821459779330685819841822787, 9.034490491297138002897845893312, 9.875694306213157314776895127814