| L(s) = 1 | + 3·3-s − 7.11·5-s + 8.88·7-s + 9·9-s + 26·11-s − 13·13-s − 21.3·15-s + 16.2·17-s + 35.5·19-s + 26.6·21-s + 153.·23-s − 74.3·25-s + 27·27-s + 223.·29-s + 126.·31-s + 78·33-s − 63.2·35-s + 217.·37-s − 39·39-s + 105.·41-s − 183.·43-s − 64.0·45-s + 96.6·47-s − 264.·49-s + 48.6·51-s + 386.·53-s − 184.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.636·5-s + 0.479·7-s + 0.333·9-s + 0.712·11-s − 0.277·13-s − 0.367·15-s + 0.231·17-s + 0.429·19-s + 0.276·21-s + 1.39·23-s − 0.595·25-s + 0.192·27-s + 1.43·29-s + 0.734·31-s + 0.411·33-s − 0.305·35-s + 0.966·37-s − 0.160·39-s + 0.403·41-s − 0.651·43-s − 0.212·45-s + 0.300·47-s − 0.769·49-s + 0.133·51-s + 1.00·53-s − 0.453·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.283355206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.283355206\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 7.11T + 125T^{2} \) |
| 7 | \( 1 - 8.88T + 343T^{2} \) |
| 11 | \( 1 - 26T + 1.33e3T^{2} \) |
| 17 | \( 1 - 16.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 105.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 96.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 386.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 34.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 93.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 741.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 640.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 182.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 288.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 963.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 481.T + 9.12e5T^{2} \) |
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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
−11.42214398917128688713621417985, −10.24788617272732909805315794294, −9.264813226358060540585988578082, −8.346252110582788825116967335954, −7.54957521691544461779378250866, −6.53715315204666169112077639816, −5.00426301943975155280014032240, −3.98523586614477440481232678366, −2.76919953794383634385098601660, −1.10022464665493037666502970190,
1.10022464665493037666502970190, 2.76919953794383634385098601660, 3.98523586614477440481232678366, 5.00426301943975155280014032240, 6.53715315204666169112077639816, 7.54957521691544461779378250866, 8.346252110582788825116967335954, 9.264813226358060540585988578082, 10.24788617272732909805315794294, 11.42214398917128688713621417985
