| L(s) = 1 | + 6.15·3-s − 18.3·5-s + 21.8·7-s + 10.8·9-s + 8.30·11-s − 53.0·13-s − 112.·15-s + 74.2·17-s + 19·19-s + 134.·21-s − 163.·23-s + 210.·25-s − 99.3·27-s + 232.·29-s + 98.4·31-s + 51.0·33-s − 399.·35-s − 296.·37-s − 326.·39-s − 434.·41-s + 171.·43-s − 198.·45-s − 366.·47-s + 134.·49-s + 456.·51-s − 138.·53-s − 152·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s − 1.63·5-s + 1.17·7-s + 0.401·9-s + 0.227·11-s − 1.13·13-s − 1.93·15-s + 1.05·17-s + 0.229·19-s + 1.39·21-s − 1.48·23-s + 1.68·25-s − 0.708·27-s + 1.48·29-s + 0.570·31-s + 0.269·33-s − 1.93·35-s − 1.31·37-s − 1.34·39-s − 1.65·41-s + 0.607·43-s − 0.657·45-s − 1.13·47-s + 0.391·49-s + 1.25·51-s − 0.359·53-s − 0.372·55-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 6.15T + 27T^{2} \) |
| 5 | \( 1 + 18.3T + 125T^{2} \) |
| 7 | \( 1 - 21.8T + 343T^{2} \) |
| 11 | \( 1 - 8.30T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 98.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 138.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 572.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 183.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 56.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 68.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 332.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 368.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 426.T + 9.12e5T^{2} \) |
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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
−8.508431743765844541609911327994, −8.139547933184006834108158182348, −7.71589192265001690673397306665, −6.84681336586784142540283497290, −5.24098438959926866780803208443, −4.43507876829310594203645225167, −3.61099934771943815120791393758, −2.76956700242838157787831580348, −1.52691469877841808685155278207, 0,
1.52691469877841808685155278207, 2.76956700242838157787831580348, 3.61099934771943815120791393758, 4.43507876829310594203645225167, 5.24098438959926866780803208443, 6.84681336586784142540283497290, 7.71589192265001690673397306665, 8.139547933184006834108158182348, 8.508431743765844541609911327994
