| L(s) = 1 | − 9.14·3-s + 17.1·5-s + 8.96·7-s + 56.7·9-s − 38.4·11-s − 40.9·13-s − 156.·15-s + 16.6·17-s − 128.·19-s − 82.0·21-s + 189.·23-s + 168.·25-s − 271.·27-s + 37.5·29-s − 176.·31-s + 351.·33-s + 153.·35-s + 114.·37-s + 374.·39-s + 63.0·41-s − 122.·43-s + 971.·45-s + 471.·47-s − 262.·49-s − 152.·51-s − 518.·53-s − 658.·55-s + ⋯ |
| L(s) = 1 | − 1.76·3-s + 1.53·5-s + 0.483·7-s + 2.10·9-s − 1.05·11-s − 0.873·13-s − 2.69·15-s + 0.237·17-s − 1.54·19-s − 0.852·21-s + 1.71·23-s + 1.34·25-s − 1.93·27-s + 0.240·29-s − 1.02·31-s + 1.85·33-s + 0.741·35-s + 0.506·37-s + 1.53·39-s + 0.240·41-s − 0.435·43-s + 3.21·45-s + 1.46·47-s − 0.765·49-s − 0.417·51-s − 1.34·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 89 | \( 1 + 89T \) |
| good | 3 | \( 1 + 9.14T + 27T^{2} \) |
| 5 | \( 1 - 17.1T + 125T^{2} \) |
| 7 | \( 1 - 8.96T + 343T^{2} \) |
| 11 | \( 1 + 38.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 37.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 63.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 518.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 106.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 435.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 894.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 814.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 414.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 589.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 489.T + 5.71e5T^{2} \) |
| 97 | \( 1 + 588.T + 9.12e5T^{2} \) |
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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
−9.913431219253213471742743871128, −8.993890630461300143318318032553, −7.56403205732864370933084149012, −6.66072518691308448386330491393, −5.87682027272927844623200167086, −5.18608555654038825227311829790, −4.63753530118609828258508580708, −2.51163314987384794134775000605, −1.40031189076496171777068371518, 0,
1.40031189076496171777068371518, 2.51163314987384794134775000605, 4.63753530118609828258508580708, 5.18608555654038825227311829790, 5.87682027272927844623200167086, 6.66072518691308448386330491393, 7.56403205732864370933084149012, 8.993890630461300143318318032553, 9.913431219253213471742743871128
