L(s) = 1 | + (1.48 + 1.01i)2-s + (1.80 + 1.67i)3-s + (0.446 + 1.13i)4-s + (−3.30 + 1.01i)5-s + (0.984 + 4.31i)6-s + (−2.12 + 1.58i)7-s + (0.311 − 1.36i)8-s + (0.229 + 3.06i)9-s + (−5.92 − 1.82i)10-s + (−0.427 + 5.70i)11-s + (−1.09 + 2.80i)12-s + (0.900 − 0.433i)13-s + (−4.74 + 0.203i)14-s + (−7.67 − 3.69i)15-s + (3.62 − 3.36i)16-s + (2.37 − 0.358i)17-s + ⋯ |
L(s) = 1 | + (1.04 + 0.714i)2-s + (1.04 + 0.967i)3-s + (0.223 + 0.568i)4-s + (−1.47 + 0.455i)5-s + (0.401 + 1.76i)6-s + (−0.801 + 0.598i)7-s + (0.109 − 0.481i)8-s + (0.0765 + 1.02i)9-s + (−1.87 − 0.578i)10-s + (−0.128 + 1.72i)11-s + (−0.317 + 0.808i)12-s + (0.249 − 0.120i)13-s + (−1.26 + 0.0542i)14-s + (−1.98 − 0.954i)15-s + (0.907 − 0.841i)16-s + (0.576 − 0.0868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.517234 + 2.38361i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.517234 + 2.38361i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.12 - 1.58i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
good | 2 | \( 1 + (-1.48 - 1.01i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.80 - 1.67i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (3.30 - 1.01i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.427 - 5.70i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (-2.37 + 0.358i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (2.59 + 4.50i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.85 - 1.03i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.83 - 3.55i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (2.85 - 4.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.54 - 9.03i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 5.70i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.24 - 9.84i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (3.94 + 2.69i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-0.0686 - 0.175i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-7.44 - 2.29i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (1.88 - 4.79i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.89 + 6.74i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.18 + 6.50i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-9.54 + 6.50i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (0.667 + 1.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.46 - 0.706i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.659 + 8.79i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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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
−10.85435654819409134486210737716, −9.953386575635856356151251510956, −9.200600717641867378346384244538, −8.253266114259917737856702646441, −7.15952786754751497028563242979, −6.73672255959128328442667095416, −5.04076282833814342464243994133, −4.48558481606413586245898893561, −3.47323200180505229030091106118, −2.94074532199675377937022839360,
0.885205996897175039945402287291, 2.65310872907550342784247017367, 3.65931414295902704912176027479, 3.85432263175784234496512456824, 5.46831658193804848794766074829, 6.69398619689368821834003713004, 7.85206889990663964499731570092, 8.200037684554518514379944004701, 9.036229883504078622127138891053, 10.65437119306986070528781719523