| L(s) = 1 | + i·3-s + 3.16i·5-s + 4.57·7-s − 9-s + 2.47i·11-s − 1.41i·13-s − 3.16·15-s − 6.47·17-s + 2.47i·19-s + 4.57i·21-s − 5.65·23-s − 5.00·25-s − i·27-s − 0.333i·29-s − 10.2·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.41i·5-s + 1.72·7-s − 0.333·9-s + 0.745i·11-s − 0.392i·13-s − 0.816·15-s − 1.56·17-s + 0.567i·19-s + 0.998i·21-s − 1.17·23-s − 1.00·25-s − 0.192i·27-s − 0.0619i·29-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.347500501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.347500501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 - 3.16iT - 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 - 2.47iT - 11T^{2} \) |
| 13 | \( 1 + 1.41iT - 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 0.333iT - 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.08iT - 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.32iT - 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 1.08T + 79T^{2} \) |
| 83 | \( 1 - 2.47iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
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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
−8.986780238969424375585120911126, −8.351687023586632043106254972247, −7.45426569636383318303426930853, −7.05790654563793873440755487481, −5.95550932228824163694244746885, −5.23347163618925579545747165390, −4.30187937775733051209202810467, −3.73403128863703321743345042540, −2.40427224538556546552747694256, −1.87676065953407738260979450268,
0.38477969875142831419531251741, 1.62052023019057126039229694760, 2.06952343583248079638531976580, 3.72286342726103646892157699386, 4.78402806821700680724125554045, 4.96600956271575364048729477211, 5.99648299095304512311078932494, 6.85604428979258257732168022966, 7.968794141130764417452636122229, 8.200222349712671578735522284282
