L(s) = 1 | + (0.5 − 0.866i)2-s + (0.758 − 2.83i)3-s + (−0.499 − 0.866i)4-s + (0.917 − 2.03i)5-s + (−2.07 − 2.07i)6-s + (−0.490 + 1.83i)7-s − 0.999·8-s + (−4.84 − 2.79i)9-s + (−1.30 − 1.81i)10-s + 0.0963i·11-s + (−2.83 + 0.758i)12-s + (1.59 + 2.75i)13-s + (1.34 + 1.34i)14-s + (−5.07 − 4.14i)15-s + (−0.5 + 0.866i)16-s + (3.73 + 2.15i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.438 − 1.63i)3-s + (−0.249 − 0.433i)4-s + (0.410 − 0.911i)5-s + (−0.846 − 0.846i)6-s + (−0.185 + 0.692i)7-s − 0.353·8-s + (−1.61 − 0.932i)9-s + (−0.413 − 0.573i)10-s + 0.0290i·11-s + (−0.817 + 0.219i)12-s + (0.441 + 0.764i)13-s + (0.358 + 0.358i)14-s + (−1.31 − 1.07i)15-s + (−0.125 + 0.216i)16-s + (0.906 + 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.406320 - 1.80589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.406320 - 1.80589i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.917 + 2.03i)T \) |
| 37 | \( 1 + (5.72 - 2.05i)T \) |
good | 3 | \( 1 + (-0.758 + 2.83i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (0.490 - 1.83i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 - 0.0963iT - 11T^{2} \) |
| 13 | \( 1 + (-1.59 - 2.75i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.73 - 2.15i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.71 - 1.26i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 29 | \( 1 + (3.50 + 3.50i)T + 29iT^{2} \) |
| 31 | \( 1 + (3.04 - 3.04i)T - 31iT^{2} \) |
| 41 | \( 1 + (-7.68 + 4.43i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 7.57T + 43T^{2} \) |
| 47 | \( 1 + (9.56 + 9.56i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.388 + 1.44i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.98 - 11.1i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-9.90 - 2.65i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (6.88 + 1.84i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.958 + 1.66i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.62 - 4.62i)T + 73iT^{2} \) |
| 79 | \( 1 + (2.89 + 0.775i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-3.32 - 12.4i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-4.69 + 1.25i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 17.6iT - 97T^{2} \) |
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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
−11.56759631507770612517027457490, −9.968335795322151899410382494809, −8.989078764551651665062326914224, −8.368741733978146588616650062733, −7.21212918310928966767547015999, −6.05910322922075029269085134556, −5.31194307648342477875989713907, −3.53303574467963588425094559025, −2.14293626902417542608089902027, −1.24309210907284898626155213856,
3.13298486655159994255646131664, 3.57050936173512782348863055662, 4.95327222481244353272047857875, 5.75549169064282964508279155073, 7.09542788149141802252934713495, 8.004174093885852776326778215426, 9.314347672930253290701587082564, 9.850569212574383979128275387175, 10.71037284764205625680334327703, 11.41281344520688988890791789817