L(s) = 1 | − 2.56·3-s + 5-s + 1.56·7-s + 3.56·9-s + 11-s − 13-s − 2.56·15-s − 2·17-s + 1.43·19-s − 4·21-s + 4.12·23-s + 25-s − 1.43·27-s − 3·29-s − 6.12·31-s − 2.56·33-s + 1.56·35-s − 8.24·37-s + 2.56·39-s − 0.876·41-s − 2·43-s + 3.56·45-s + 1.12·47-s − 4.56·49-s + 5.12·51-s + 2·53-s + 55-s + ⋯ |
L(s) = 1 | − 1.47·3-s + 0.447·5-s + 0.590·7-s + 1.18·9-s + 0.301·11-s − 0.277·13-s − 0.661·15-s − 0.485·17-s + 0.330·19-s − 0.872·21-s + 0.859·23-s + 0.200·25-s − 0.276·27-s − 0.557·29-s − 1.09·31-s − 0.445·33-s + 0.263·35-s − 1.35·37-s + 0.410·39-s − 0.136·41-s − 0.304·43-s + 0.530·45-s + 0.163·47-s − 0.651·49-s + 0.717·51-s + 0.274·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 0.876T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 9.24T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−7.42661560122772483318352343471, −6.92013750737318921353389405276, −6.24530192991694714520287515472, −5.42971363086014532713759134556, −5.10927441128920111702496012987, −4.33442611046960920993616809952, −3.30360802228921267599748524781, −2.03965610050917304495721583321, −1.21436722948717851252929564920, 0,
1.21436722948717851252929564920, 2.03965610050917304495721583321, 3.30360802228921267599748524781, 4.33442611046960920993616809952, 5.10927441128920111702496012987, 5.42971363086014532713759134556, 6.24530192991694714520287515472, 6.92013750737318921353389405276, 7.42661560122772483318352343471