| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 18.4·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 36.8·10-s + 34.5·11-s − 12·12-s − 64.3·13-s − 14·14-s − 55.3·15-s + 16·16-s + 55.4·17-s + 18·18-s + 52.9·19-s + 73.7·20-s + 21·21-s + 69.0·22-s + 23·23-s − 24·24-s + 214.·25-s − 128.·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.64·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.16·10-s + 0.946·11-s − 0.288·12-s − 1.37·13-s − 0.267·14-s − 0.952·15-s + 0.250·16-s + 0.791·17-s + 0.235·18-s + 0.639·19-s + 0.824·20-s + 0.218·21-s + 0.669·22-s + 0.208·23-s − 0.204·24-s + 1.71·25-s − 0.970·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.952581314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.952581314\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 18.4T + 125T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 9.74T + 2.43e4T^{2} \) |
| 31 | \( 1 - 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 17.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 359.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 420.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 413.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 13.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 72.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 337.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 686.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 513.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 27.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 477.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−9.759935149208274721655995725399, −9.208830450730273080089642411701, −7.64914691878359140069517417900, −6.70898054599149121558668161104, −6.11350612817867634803870976062, −5.35052176236109079226812823613, −4.61362040417802234149343191805, −3.20247931623055934840303868845, −2.16356011852314180800175670424, −1.05058638914229990971665325452,
1.05058638914229990971665325452, 2.16356011852314180800175670424, 3.20247931623055934840303868845, 4.61362040417802234149343191805, 5.35052176236109079226812823613, 6.11350612817867634803870976062, 6.70898054599149121558668161104, 7.64914691878359140069517417900, 9.208830450730273080089642411701, 9.759935149208274721655995725399
