L(s) = 1 | + (−0.359 − 1.10i)2-s + (0.525 − 0.381i)4-s + (0.309 − 0.951i)5-s + (3.46 − 2.51i)7-s + (−2.49 − 1.80i)8-s − 1.16·10-s + (3.15 − 1.00i)11-s + (1.59 + 4.91i)13-s + (−4.03 − 2.92i)14-s + (−0.704 + 2.16i)16-s + (−1.54 + 4.75i)17-s + (−4.53 − 3.29i)19-s + (−0.200 − 0.617i)20-s + (−2.25 − 3.12i)22-s + 0.219·23-s + ⋯ |
L(s) = 1 | + (−0.253 − 0.781i)2-s + (0.262 − 0.190i)4-s + (0.138 − 0.425i)5-s + (1.31 − 0.952i)7-s + (−0.880 − 0.639i)8-s − 0.367·10-s + (0.952 − 0.304i)11-s + (0.442 + 1.36i)13-s + (−1.07 − 0.782i)14-s + (−0.176 + 0.541i)16-s + (−0.374 + 1.15i)17-s + (−1.03 − 0.755i)19-s + (−0.0448 − 0.138i)20-s + (−0.479 − 0.667i)22-s + 0.0458·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.910996 - 1.32285i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.910996 - 1.32285i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (-3.15 + 1.00i)T \) |
good | 2 | \( 1 + (0.359 + 1.10i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + (-3.46 + 2.51i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 4.91i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.54 - 4.75i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.53 + 3.29i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 0.219T + 23T^{2} \) |
| 29 | \( 1 + (-5.19 + 3.77i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.874 + 2.69i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.17 - 2.30i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.74 - 3.44i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 + (0.192 + 0.139i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.783 + 2.41i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.36 - 4.62i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.50 - 4.62i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + (-3.08 + 9.49i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (11.7 - 8.55i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.47 - 7.61i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.87 - 11.9i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 2.56T + 89T^{2} \) |
| 97 | \( 1 + (-0.621 - 1.91i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.91342934192471589123163546198, −9.959970540517548038463099651359, −8.923603465655913562624512770419, −8.293918239690006259286770792737, −6.85229673340528964196116655577, −6.21077935707542024525064951398, −4.57293984215362689039624955035, −3.90945465482265295478297833432, −2.02118384854476781012478782730, −1.20002819559178525208723077006,
1.95867362279268281919616499328, 3.18727888059829190936647715776, 4.86852968654595641876567751191, 5.80212327561604013059904591487, 6.66847160948610476403318172424, 7.64855942917577827550974541890, 8.454474283748839425855556720946, 9.018800292033856174772118946454, 10.42068487302077203449640616264, 11.28011248903383336603920915066