| L(s) = 1 | + 5.24·2-s + 7.24·3-s + 19.5·4-s + 5·5-s + 37.9·6-s + 24.5·7-s + 60.3·8-s + 25.4·9-s + 26.2·10-s − 11·11-s + 141.·12-s − 73.2·13-s + 128.·14-s + 36.2·15-s + 160.·16-s − 54.9·17-s + 133.·18-s + 19·19-s + 97.5·20-s + 177.·21-s − 57.6·22-s − 165.·23-s + 437.·24-s + 25·25-s − 384.·26-s − 10.9·27-s + 478.·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 1.39·3-s + 2.43·4-s + 0.447·5-s + 2.58·6-s + 1.32·7-s + 2.66·8-s + 0.944·9-s + 0.829·10-s − 0.301·11-s + 3.39·12-s − 1.56·13-s + 2.45·14-s + 0.623·15-s + 2.50·16-s − 0.784·17-s + 1.75·18-s + 0.229·19-s + 1.09·20-s + 1.84·21-s − 0.559·22-s − 1.50·23-s + 3.71·24-s + 0.200·25-s − 2.89·26-s − 0.0780·27-s + 3.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.89803209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.89803209\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 5.24T + 8T^{2} \) |
| 3 | \( 1 - 7.24T + 27T^{2} \) |
| 7 | \( 1 - 24.5T + 343T^{2} \) |
| 13 | \( 1 + 73.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 77.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 339.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 256.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 275.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 418.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 332.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 641.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 644.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 700.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.39e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 695.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 627.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−9.689124553049591676509526541163, −8.329979469979762931342178086143, −7.83846481292653933068252678801, −6.93517414740674387295740804654, −5.90358678113742444545431222639, −4.71758189028383426582842332762, −4.54250735083232084946640951298, −3.17832987012015599192851175213, −2.37231050375514608020980674242, −1.84313595931806088596141542628,
1.84313595931806088596141542628, 2.37231050375514608020980674242, 3.17832987012015599192851175213, 4.54250735083232084946640951298, 4.71758189028383426582842332762, 5.90358678113742444545431222639, 6.93517414740674387295740804654, 7.83846481292653933068252678801, 8.329979469979762931342178086143, 9.689124553049591676509526541163
