| L(s) = 1 | − 1.32·2-s − 6.23·4-s + 5·5-s − 24.1·7-s + 18.8·8-s − 6.63·10-s + 8.27·11-s + 87.1·13-s + 31.9·14-s + 24.8·16-s + 51.9·17-s − 88.5·19-s − 31.1·20-s − 10.9·22-s − 129.·23-s + 25·25-s − 115.·26-s + 150.·28-s − 271.·29-s + 224.·31-s − 184.·32-s − 68.8·34-s − 120.·35-s − 70.5·37-s + 117.·38-s + 94.4·40-s + 366.·41-s + ⋯ |
| L(s) = 1 | − 0.469·2-s − 0.779·4-s + 0.447·5-s − 1.30·7-s + 0.834·8-s − 0.209·10-s + 0.226·11-s + 1.85·13-s + 0.610·14-s + 0.388·16-s + 0.740·17-s − 1.06·19-s − 0.348·20-s − 0.106·22-s − 1.17·23-s + 0.200·25-s − 0.871·26-s + 1.01·28-s − 1.73·29-s + 1.30·31-s − 1.01·32-s − 0.347·34-s − 0.582·35-s − 0.313·37-s + 0.501·38-s + 0.373·40-s + 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 1.32T + 8T^{2} \) |
| 7 | \( 1 + 24.1T + 343T^{2} \) |
| 11 | \( 1 - 8.27T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 271.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 224.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 366.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 195.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 359.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 29.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 858.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 41.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 549.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 185.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 80.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 576.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 224.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 555.T + 9.12e5T^{2} \) |
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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
−10.16478140631094759944993060851, −9.459405219447065759895262686290, −8.726889212965877575502493346170, −7.82325278058687865780446799668, −6.37039664675292635470370550324, −5.83845888550232078617606521778, −4.22897346335225905324725636687, −3.34437935696238541964785295487, −1.47157720848357479395799375154, 0,
1.47157720848357479395799375154, 3.34437935696238541964785295487, 4.22897346335225905324725636687, 5.83845888550232078617606521778, 6.37039664675292635470370550324, 7.82325278058687865780446799668, 8.726889212965877575502493346170, 9.459405219447065759895262686290, 10.16478140631094759944993060851
