L(s) = 1 | + 2-s + (1.69 + 0.341i)3-s + 4-s + (1.23 − 1.86i)5-s + (1.69 + 0.341i)6-s − 0.843i·7-s + 8-s + (2.76 + 1.16i)9-s + (1.23 − 1.86i)10-s − 2.73·11-s + (1.69 + 0.341i)12-s + 1.61·13-s − 0.843i·14-s + (2.74 − 2.73i)15-s + 16-s + 3.82i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.980 + 0.197i)3-s + 0.5·4-s + (0.554 − 0.832i)5-s + (0.693 + 0.139i)6-s − 0.318i·7-s + 0.353·8-s + (0.922 + 0.387i)9-s + (0.392 − 0.588i)10-s − 0.824·11-s + (0.490 + 0.0987i)12-s + 0.448·13-s − 0.225i·14-s + (0.707 − 0.706i)15-s + 0.250·16-s + 0.927i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.59847 - 0.550616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.59847 - 0.550616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.69 - 0.341i)T \) |
| 5 | \( 1 + (-1.23 + 1.86i)T \) |
| 31 | \( 1 + (-2.58 - 4.93i)T \) |
good | 7 | \( 1 + 0.843iT - 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 - 3.82iT - 17T^{2} \) |
| 19 | \( 1 + 0.779T + 19T^{2} \) |
| 23 | \( 1 + 3.42iT - 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 37 | \( 1 - 4.77T + 37T^{2} \) |
| 41 | \( 1 + 7.71iT - 41T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 1.15iT - 53T^{2} \) |
| 59 | \( 1 - 3.35iT - 59T^{2} \) |
| 61 | \( 1 - 15.3iT - 61T^{2} \) |
| 67 | \( 1 + 8.81iT - 67T^{2} \) |
| 71 | \( 1 - 13.6iT - 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 4.05iT - 79T^{2} \) |
| 83 | \( 1 + 8.96iT - 83T^{2} \) |
| 89 | \( 1 - 4.16T + 89T^{2} \) |
| 97 | \( 1 - 0.560iT - 97T^{2} \) |
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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
−10.24666256926741407700229649321, −9.038126367301191606742239809643, −8.408046652506675864988489443403, −7.61871209719507411100725272000, −6.50950337185104499644045297513, −5.49547362619638900318572438345, −4.59351596798787577395003539922, −3.78336748127653419461121807694, −2.60681195378565219540602375653, −1.53051592666516648154349317544,
1.85343991564110460467345537691, 2.78196333480990104638056839180, 3.47551800693550432264063667873, 4.76946544554067386851179676221, 5.83099747185864602724830789484, 6.68005606535366355286789766337, 7.53429067242062377899693945348, 8.239758457594035206880044711166, 9.493902239396030849204249095732, 9.923221704040701079031556537420