| L(s) = 1 | + 7.74·3-s + 1.78·5-s + 26.5·7-s + 33.0·9-s + 59.3·11-s + 45.5·13-s + 13.8·15-s − 84.3·17-s + 58.1·19-s + 206.·21-s − 125.·23-s − 121.·25-s + 46.9·27-s + 7.96·29-s − 225.·31-s + 460.·33-s + 47.5·35-s − 37·37-s + 353.·39-s + 207.·41-s − 453.·43-s + 59.0·45-s + 253.·47-s + 364.·49-s − 653.·51-s + 508.·53-s + 106.·55-s + ⋯ |
| L(s) = 1 | + 1.49·3-s + 0.159·5-s + 1.43·7-s + 1.22·9-s + 1.62·11-s + 0.972·13-s + 0.238·15-s − 1.20·17-s + 0.702·19-s + 2.14·21-s − 1.13·23-s − 0.974·25-s + 0.334·27-s + 0.0510·29-s − 1.30·31-s + 2.42·33-s + 0.229·35-s − 0.164·37-s + 1.44·39-s + 0.791·41-s − 1.60·43-s + 0.195·45-s + 0.785·47-s + 1.06·49-s − 1.79·51-s + 1.31·53-s + 0.260·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.512159230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.512159230\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + 37T \) |
| good | 3 | \( 1 - 7.74T + 27T^{2} \) |
| 5 | \( 1 - 1.78T + 125T^{2} \) |
| 7 | \( 1 - 26.5T + 343T^{2} \) |
| 11 | \( 1 - 59.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 7.96T + 2.43e4T^{2} \) |
| 31 | \( 1 + 225.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 207.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 453.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 253.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 508.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 26.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 709.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 468.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 117.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 701.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 13.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.82e3T + 9.12e5T^{2} \) |
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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
−10.06139220678810079193779671765, −9.053849150068521631021609421401, −8.658546939392395910515720968884, −7.86438155307851788548390802816, −6.92432505979797160902052647077, −5.70283149500375362130387054111, −4.22623327342723240341488290063, −3.71016024712934126481688265206, −2.13155178742416536540852199469, −1.45309752090454566014190881391,
1.45309752090454566014190881391, 2.13155178742416536540852199469, 3.71016024712934126481688265206, 4.22623327342723240341488290063, 5.70283149500375362130387054111, 6.92432505979797160902052647077, 7.86438155307851788548390802816, 8.658546939392395910515720968884, 9.053849150068521631021609421401, 10.06139220678810079193779671765
