| L(s) = 1 | − 2·2-s − 8.33·3-s + 4·4-s + 16.6·6-s − 8·8-s + 42.5·9-s − 26.2·11-s − 33.3·12-s − 38.1·13-s + 16·16-s + 2.39·17-s − 85.1·18-s − 99.7·19-s + 52.5·22-s − 101.·23-s + 66.7·24-s + 76.3·26-s − 129.·27-s − 223.·29-s + 267.·31-s − 32·32-s + 219.·33-s − 4.78·34-s + 170.·36-s − 363.·37-s + 199.·38-s + 318.·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.60·3-s + 0.5·4-s + 1.13·6-s − 0.353·8-s + 1.57·9-s − 0.720·11-s − 0.802·12-s − 0.814·13-s + 0.250·16-s + 0.0341·17-s − 1.11·18-s − 1.20·19-s + 0.509·22-s − 0.922·23-s + 0.567·24-s + 0.575·26-s − 0.924·27-s − 1.43·29-s + 1.54·31-s − 0.176·32-s + 1.15·33-s − 0.0241·34-s + 0.788·36-s − 1.61·37-s + 0.852·38-s + 1.30·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.005367191027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005367191027\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 8.33T + 27T^{2} \) |
| 11 | \( 1 + 26.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.39T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 363.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 94.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 406.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 543.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 393.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 504.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 54.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 889.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 525.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.65e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 5.76T + 9.12e5T^{2} \) |
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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
−8.440910824564132496075675803987, −7.86315132120379688666229230276, −6.81854771841214437811116758314, −6.46607530840959081223208675225, −5.45665034105279727878880251264, −4.96812628870072152064962713953, −3.89831418524029806104462846426, −2.47379560279565142387594740385, −1.48843484124618434516322952893, −0.03954458127911617588126787226,
0.03954458127911617588126787226, 1.48843484124618434516322952893, 2.47379560279565142387594740385, 3.89831418524029806104462846426, 4.96812628870072152064962713953, 5.45665034105279727878880251264, 6.46607530840959081223208675225, 6.81854771841214437811116758314, 7.86315132120379688666229230276, 8.440910824564132496075675803987
