L(s) = 1 | + (0.5 − 0.866i)2-s + (0.304 − 1.13i)3-s + (−0.499 − 0.866i)4-s + (2.21 − 0.336i)5-s + (−0.832 − 0.832i)6-s + (0.993 − 3.70i)7-s − 0.999·8-s + (1.39 + 0.806i)9-s + (0.814 − 2.08i)10-s + 5.46i·11-s + (−1.13 + 0.304i)12-s + (0.988 + 1.71i)13-s + (−2.71 − 2.71i)14-s + (0.291 − 2.61i)15-s + (−0.5 + 0.866i)16-s + (−6.08 − 3.51i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.175 − 0.656i)3-s + (−0.249 − 0.433i)4-s + (0.988 − 0.150i)5-s + (−0.339 − 0.339i)6-s + (0.375 − 1.40i)7-s − 0.353·8-s + (0.465 + 0.268i)9-s + (0.257 − 0.658i)10-s + 1.64i·11-s + (−0.328 + 0.0879i)12-s + (0.274 + 0.475i)13-s + (−0.725 − 0.725i)14-s + (0.0751 − 0.675i)15-s + (−0.125 + 0.216i)16-s + (−1.47 − 0.852i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27565 - 1.50980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27565 - 1.50980i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-2.21 + 0.336i)T \) |
| 37 | \( 1 + (-4.05 - 4.53i)T \) |
good | 3 | \( 1 + (-0.304 + 1.13i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-0.993 + 3.70i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 - 5.46iT - 11T^{2} \) |
| 13 | \( 1 + (-0.988 - 1.71i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (6.08 + 3.51i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.64 + 0.440i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 6.08T + 23T^{2} \) |
| 29 | \( 1 + (-4.74 - 4.74i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.415 - 0.415i)T - 31iT^{2} \) |
| 41 | \( 1 + (0.895 - 0.517i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 8.46T + 43T^{2} \) |
| 47 | \( 1 + (4.67 + 4.67i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.67 - 9.99i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.818 + 3.05i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.58 - 0.692i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-9.46 - 2.53i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.59 - 4.48i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.88 + 2.88i)T + 73iT^{2} \) |
| 79 | \( 1 + (-5.43 - 1.45i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-2.80 - 10.4i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (8.91 - 2.38i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 3.18iT - 97T^{2} \) |
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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
−11.09743423821875084620701013142, −10.17905473334670925037724947552, −9.699702607177436460942217004922, −8.371558357562571097840001750394, −7.04137664515932351831592817827, −6.65878556130951594501181524150, −4.81215892617135270980780670229, −4.31599412291616453435211796384, −2.28667940533709440360008189962, −1.46836172998241924559133132475,
2.28700414487956240838492150275, 3.63616240901118086025894279045, 4.93402810472766241980240394832, 6.02522370896386044444442448128, 6.34250533649582252868186085801, 8.317998089156411717704974658368, 8.683725477987912602239353485783, 9.668105463406352193597882663507, 10.67551773077037243039768938511, 11.61950593723437110813025394604