| L(s) = 1 | − 4.73·3-s + 18.0·5-s + 0.213·7-s − 4.58·9-s + 3.39·11-s + 90.7·13-s − 85.5·15-s − 2.59·17-s − 19·19-s − 1.01·21-s + 26.6·23-s + 201.·25-s + 149.·27-s − 60.1·29-s − 176.·31-s − 16.0·33-s + 3.85·35-s + 154.·37-s − 429.·39-s + 434.·41-s + 365.·43-s − 82.8·45-s + 204.·47-s − 342.·49-s + 12.2·51-s + 135.·53-s + 61.3·55-s + ⋯ |
| L(s) = 1 | − 0.911·3-s + 1.61·5-s + 0.0115·7-s − 0.169·9-s + 0.0930·11-s + 1.93·13-s − 1.47·15-s − 0.0370·17-s − 0.229·19-s − 0.0104·21-s + 0.241·23-s + 1.61·25-s + 1.06·27-s − 0.384·29-s − 1.02·31-s − 0.0847·33-s + 0.0186·35-s + 0.684·37-s − 1.76·39-s + 1.65·41-s + 1.29·43-s − 0.274·45-s + 0.633·47-s − 0.999·49-s + 0.0337·51-s + 0.351·53-s + 0.150·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.304550162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304550162\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 4.73T + 27T^{2} \) |
| 5 | \( 1 - 18.0T + 125T^{2} \) |
| 7 | \( 1 - 0.213T + 343T^{2} \) |
| 11 | \( 1 - 3.39T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.59T + 4.91e3T^{2} \) |
| 23 | \( 1 - 26.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 759.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 284.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 590.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 972.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 204.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 782.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 213.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−9.170331138499425835494239592240, −8.954449949886780277907087683705, −7.63769198565172058046091780666, −6.34550811479308048082527021508, −6.04492715214467868858643370953, −5.48594760382382128646056107081, −4.33957174762136568794719438689, −3.02765136271079751466516005713, −1.80182981627755068622544222603, −0.855572303946344087239671871350,
0.855572303946344087239671871350, 1.80182981627755068622544222603, 3.02765136271079751466516005713, 4.33957174762136568794719438689, 5.48594760382382128646056107081, 6.04492715214467868858643370953, 6.34550811479308048082527021508, 7.63769198565172058046091780666, 8.954449949886780277907087683705, 9.170331138499425835494239592240
