| L(s) = 1 | + 4i·3-s + 16i·7-s + 11·9-s + 60·11-s − 86i·13-s + 18i·17-s + 44·19-s − 64·21-s + 48i·23-s + 152i·27-s + 186·29-s − 176·31-s + 240i·33-s + 254i·37-s + 344·39-s + ⋯ |
| L(s) = 1 | + 0.769i·3-s + 0.863i·7-s + 0.407·9-s + 1.64·11-s − 1.83i·13-s + 0.256i·17-s + 0.531·19-s − 0.665·21-s + 0.435i·23-s + 1.08i·27-s + 1.19·29-s − 1.01·31-s + 1.26i·33-s + 1.12i·37-s + 1.41·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.238190706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238190706\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4iT - 27T^{2} \) |
| 7 | \( 1 - 16iT - 343T^{2} \) |
| 11 | \( 1 - 60T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 44T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 186T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176T + 2.97e4T^{2} \) |
| 37 | \( 1 - 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 186T + 6.89e4T^{2} \) |
| 43 | \( 1 + 100iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 168iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 498iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 252T + 2.05e5T^{2} \) |
| 61 | \( 1 + 58T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 168T + 3.57e5T^{2} \) |
| 73 | \( 1 + 506iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 272T + 4.93e5T^{2} \) |
| 83 | \( 1 - 948iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 766iT - 9.12e5T^{2} \) |
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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
−10.88553657362914629775673542227, −10.03757263700532109462940454548, −9.278612194243327085820332517192, −8.470641273346830992127916313401, −7.28953775353415661464669081387, −6.07069903855916543696095651768, −5.19424067709232827279617385093, −4.00878297577036044080831815760, −3.00463255664906067961686268508, −1.22841051218494087435189730145,
0.948532497873303986628458095219, 1.91250631198220828678566325703, 3.81115020879221736491172020123, 4.54473166964225574307258386249, 6.37189256320536165246887805706, 6.85758040629342503314450057238, 7.63043403653698445783749631998, 9.003198762170128608930132210084, 9.595669822183827216900679332762, 10.83168962650444378522874956162
