L(s) = 1 | + 2i·3-s − 3i·7-s − 9-s − 3·11-s + 4i·13-s + 5i·17-s + 19-s + 6·21-s + 4i·27-s − 2·29-s + 8·31-s − 6i·33-s − 10i·37-s − 8·39-s + 6·41-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 1.13i·7-s − 0.333·9-s − 0.904·11-s + 1.10i·13-s + 1.21i·17-s + 0.229·19-s + 1.30·21-s + 0.769i·27-s − 0.371·29-s + 1.43·31-s − 1.04i·33-s − 1.64i·37-s − 1.28·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.178716505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178716505\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 7iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 15iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−9.030809037733862156623656225545, −8.025698111366362996554971981534, −7.44808859959028274199562971376, −6.59216034850201340661285811958, −5.73860933803634296000800871830, −4.79527442298231450527328722980, −4.17522762493071544664550711140, −3.73458939752118664797158719667, −2.56870775710239608364879151050, −1.28912794278658653083113113004,
0.35376852483011799093684954549, 1.53455697870083403354735221838, 2.73670327717184577594268208890, 2.90207909485338735430160526495, 4.59814527797673581780685692576, 5.33072368917676010425119398480, 6.00369101759861900478797809969, 6.71048135358289171715399657223, 7.68841256722481721106082275195, 7.904255794823810826081156019125