| L(s) = 1 | + (−0.825 + 1.52i)3-s + 3.94i·5-s + (−1.63 − 2.51i)9-s + 3.52·11-s + 2·13-s + (−6.00 − 3.25i)15-s + 5.02i·17-s + 3.04i·19-s + 1.65·23-s − 10.5·25-s + (5.17 − 0.418i)27-s + 6.80i·29-s + 1.73i·31-s + (−2.91 + 5.37i)33-s + 1.27·37-s + ⋯ |
| L(s) = 1 | + (−0.476 + 0.879i)3-s + 1.76i·5-s + (−0.545 − 0.837i)9-s + 1.06·11-s + 0.554·13-s + (−1.55 − 0.840i)15-s + 1.21i·17-s + 0.698i·19-s + 0.344·23-s − 2.10·25-s + (0.996 − 0.0805i)27-s + 1.26i·29-s + 0.311i·31-s + (−0.506 + 0.935i)33-s + 0.209·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.369562138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369562138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.825 - 1.52i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.94iT - 5T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.02iT - 17T^{2} \) |
| 19 | \( 1 - 3.04iT - 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 - 6.80iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 - 2.16iT - 41T^{2} \) |
| 43 | \( 1 - 0.837iT - 43T^{2} \) |
| 47 | \( 1 + 4.95T + 47T^{2} \) |
| 53 | \( 1 - 9.66iT - 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 16.0iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 + 7.19iT - 89T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−9.499111262029258874683666571892, −8.760079340302788059964764811690, −7.78295279359963004977776926148, −6.72413093960018256979310452939, −6.35122977368902113474489519693, −5.66141701487622584959344193940, −4.38827197950402607992702059745, −3.57509082069159517739135763014, −3.11966522623181070840271630150, −1.60943426078595621445160800176,
0.55833128485709130910077514514, 1.24757270643941948072164164676, 2.37225419229712614514358576783, 3.89810496769573762066433373480, 4.79148362150749023284410316559, 5.38895963546146766414600621244, 6.24923362659605844421725633935, 7.01581921831556212224057616294, 7.913245002498068443475644604012, 8.594996428927645820319860660022
