| L(s) = 1 | + 2.73i·3-s − 4.73·7-s − 4.46·9-s − 3.46i·11-s − 3.46i·13-s + 3.46·17-s + 2i·19-s − 12.9i·21-s + 2.19·23-s − 3.99i·27-s − 2.53·31-s + 9.46·33-s − 6i·37-s + 9.46·39-s + 9.46·41-s + ⋯ |
| L(s) = 1 | + 1.57i·3-s − 1.78·7-s − 1.48·9-s − 1.04i·11-s − 0.960i·13-s + 0.840·17-s + 0.458i·19-s − 2.82i·21-s + 0.457·23-s − 0.769i·27-s − 0.455·31-s + 1.64·33-s − 0.986i·37-s + 1.51·39-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9265238009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9265238009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.73iT - 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 0.196iT - 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 0.928iT - 61T^{2} \) |
| 67 | \( 1 + 0.196iT - 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 1.26iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−9.605216743529937111612956283437, −8.864319150101904333975502239548, −7.994397253426378401997303545378, −6.81990609907789252638403494659, −5.73933794685541887185471948824, −5.49531226044552528410680903260, −4.07990890101526082200394947437, −3.38813411889109783797131683705, −2.92334236876681849653032330549, −0.41373076572827071569836634462,
1.08704178345478849237219758118, 2.30860160139900247381029996398, 3.13632651833736111570493731919, 4.36142487485072428769078152476, 5.74594863772436518834234912207, 6.42063915075709875881832336038, 7.10456063052636247221000063192, 7.43288960841518634693341682025, 8.628010365579365389778493058430, 9.457949193015892121991728841652
