| L(s) = 1 | + 7.88·3-s − 14.0·5-s − 34.0·7-s + 35.2·9-s + 52.3·11-s − 64.2·13-s − 110.·15-s − 109.·17-s − 72.2·19-s − 268.·21-s + 98.4·23-s + 71.1·25-s + 64.8·27-s − 29·29-s + 26.6·31-s + 412.·33-s + 477.·35-s − 147.·37-s − 507.·39-s + 134.·41-s − 69.3·43-s − 493.·45-s − 235.·47-s + 818.·49-s − 866.·51-s + 83.8·53-s − 733.·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 1.25·5-s − 1.84·7-s + 1.30·9-s + 1.43·11-s − 1.37·13-s − 1.90·15-s − 1.56·17-s − 0.872·19-s − 2.79·21-s + 0.892·23-s + 0.569·25-s + 0.462·27-s − 0.185·29-s + 0.154·31-s + 2.17·33-s + 2.30·35-s − 0.654·37-s − 2.08·39-s + 0.510·41-s − 0.245·43-s − 1.63·45-s − 0.731·47-s + 2.38·49-s − 2.38·51-s + 0.217·53-s − 1.79·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 7.88T + 27T^{2} \) |
| 5 | \( 1 + 14.0T + 125T^{2} \) |
| 7 | \( 1 + 34.0T + 343T^{2} \) |
| 11 | \( 1 - 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.4T + 1.21e4T^{2} \) |
| 31 | \( 1 - 26.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 134.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 69.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 83.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 164.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 916.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 34.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 827.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 360.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 291.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 472.T + 9.12e5T^{2} \) |
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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
−11.37777964352163382818414286144, −9.899535288171813381914695814056, −9.168086433881538300653600254565, −8.511539961735972983977261172211, −7.16541973288646217842766049983, −6.67510143728699757940854467102, −4.27940224045732902416928477174, −3.55352137787283329290729638923, −2.49761884886699164232881243738, 0,
2.49761884886699164232881243738, 3.55352137787283329290729638923, 4.27940224045732902416928477174, 6.67510143728699757940854467102, 7.16541973288646217842766049983, 8.511539961735972983977261172211, 9.168086433881538300653600254565, 9.899535288171813381914695814056, 11.37777964352163382818414286144
