| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.094620751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.094620751\) |
| \(L(\frac{3}{2})\) |
|
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
−9.672765019090179293699409862668, −8.844193062644247438858209206561, −8.288659212238985979386897896395, −7.51426980500117925683936107390, −6.03671401963567622367475751265, −5.37370544911769546755863656317, −4.50777750953115654925890762381, −4.02682533207793405846329703041, −2.87733466202939729634501969323, −1.41889562318520764193328000033,
0.927734229515040444581433338522, 1.82388916887065439124114505538, 2.63921646402657013464883227086, 4.24573980943864695091015876783, 5.22193474273691020405312855045, 5.91537943091034196527328501883, 7.07472698479222963329002625533, 7.66000115365681136170800493881, 8.011242452582814439626125667834, 8.858293624226368296821700489465
