| L(s) = 1 | + 0.607·2-s − 7.63·4-s − 5·5-s + 8.95·7-s − 9.49·8-s − 3.03·10-s + 11·11-s − 0.460·13-s + 5.43·14-s + 55.2·16-s − 128.·17-s − 0.0245·19-s + 38.1·20-s + 6.67·22-s − 171.·23-s + 25·25-s − 0.279·26-s − 68.3·28-s + 226.·29-s + 195.·31-s + 109.·32-s − 77.9·34-s − 44.7·35-s + 338.·37-s − 0.0149·38-s + 47.4·40-s − 136.·41-s + ⋯ |
| L(s) = 1 | + 0.214·2-s − 0.953·4-s − 0.447·5-s + 0.483·7-s − 0.419·8-s − 0.0960·10-s + 0.301·11-s − 0.00982·13-s + 0.103·14-s + 0.863·16-s − 1.83·17-s − 0.000296·19-s + 0.426·20-s + 0.0647·22-s − 1.55·23-s + 0.200·25-s − 0.00210·26-s − 0.461·28-s + 1.45·29-s + 1.13·31-s + 0.604·32-s − 0.393·34-s − 0.216·35-s + 1.50·37-s − 6.37e − 5·38-s + 0.187·40-s − 0.521·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.369649616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369649616\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 0.607T + 8T^{2} \) |
| 7 | \( 1 - 8.95T + 343T^{2} \) |
| 13 | \( 1 + 0.460T + 2.19e3T^{2} \) |
| 17 | \( 1 + 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.0245T + 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 622.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 9.86T + 2.05e5T^{2} \) |
| 61 | \( 1 - 902.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 146.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 893.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 459.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 125.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 150.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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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
−10.51077584795064688855209587308, −9.589973765481652019236556982948, −8.591925547094893204402414074136, −8.121276013650797285026179413688, −6.80930888106957094921441430802, −5.76755780759137923603263040858, −4.46606958221167641800799077777, −4.13058111060786700180640580822, −2.49386187125924133385062087394, −0.70836904875813104422309919805,
0.70836904875813104422309919805, 2.49386187125924133385062087394, 4.13058111060786700180640580822, 4.46606958221167641800799077777, 5.76755780759137923603263040858, 6.80930888106957094921441430802, 8.121276013650797285026179413688, 8.591925547094893204402414074136, 9.589973765481652019236556982948, 10.51077584795064688855209587308
