| L(s) = 1 | + i·2-s − 4-s − 1.74·5-s + 7-s − i·8-s − 1.74i·10-s + 2.95i·11-s + 5.32·13-s + i·14-s + 16-s − 6.27i·17-s + 4.95i·19-s + 1.74·20-s − 2.95·22-s + 4.95·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.780·5-s + 0.377·7-s − 0.353i·8-s − 0.551i·10-s + 0.891i·11-s + 1.47·13-s + 0.267i·14-s + 0.250·16-s − 1.52i·17-s + 1.13i·19-s + 0.390·20-s − 0.630·22-s + 1.03·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596407394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596407394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 + (3.21 + 4.32i)T \) |
| good | 5 | \( 1 + 1.74T + 5T^{2} \) |
| 11 | \( 1 - 2.95iT - 11T^{2} \) |
| 13 | \( 1 - 5.32T + 13T^{2} \) |
| 17 | \( 1 + 6.27iT - 17T^{2} \) |
| 19 | \( 1 - 4.95iT - 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 31 | \( 1 + 3.21iT - 31T^{2} \) |
| 37 | \( 1 + 8.44iT - 37T^{2} \) |
| 41 | \( 1 + 4.27iT - 41T^{2} \) |
| 43 | \( 1 + 2.64iT - 43T^{2} \) |
| 47 | \( 1 - 13.5iT - 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 7.32T + 59T^{2} \) |
| 61 | \( 1 + 7.46iT - 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 - 8.44T + 71T^{2} \) |
| 73 | \( 1 + 5.85iT - 73T^{2} \) |
| 79 | \( 1 - 2.64iT - 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 - 2.36iT - 89T^{2} \) |
| 97 | \( 1 - 3.63iT - 97T^{2} \) |
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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
−8.405105349936390175268960101340, −7.70946975441327850398100822353, −7.37278383143240049086126996112, −6.42209012046761957848377500152, −5.66810682761434009427890408743, −4.83629768835425214389978603812, −4.06848441072555231598506420974, −3.41252492480445306864716102132, −2.01832983160290841609484459981, −0.69381224564854372832712965890,
0.841969962653398602224178881293, 1.75261400454371125445332906038, 3.19241928798967655892347514556, 3.57497675123550365120845171126, 4.46228072824432986805656365331, 5.31988703895113529860133995418, 6.20375277534287285949991063583, 6.98097018896827655986274550962, 8.091692167526985845190850794263, 8.508507514951151186368334717472
