| L(s) = 1 | + (2.73 − 0.732i)2-s + (9.05 − 5.22i)3-s + (6.92 − 4i)4-s + (8.32 + 2.22i)5-s + (20.9 − 20.9i)6-s + (41.9 + 72.7i)7-s + (15.9 − 16i)8-s + (14.1 − 24.4i)9-s + 24.3·10-s − 207. i·11-s + (41.8 − 72.4i)12-s + (−148. − 39.8i)13-s + (167. + 167. i)14-s + (86.9 − 23.3i)15-s + (31.9 − 55.4i)16-s + (65.3 + 243. i)17-s + ⋯ |
| L(s) = 1 | + (0.683 − 0.183i)2-s + (1.00 − 0.580i)3-s + (0.433 − 0.250i)4-s + (0.332 + 0.0891i)5-s + (0.580 − 0.580i)6-s + (0.856 + 1.48i)7-s + (0.249 − 0.250i)8-s + (0.174 − 0.302i)9-s + 0.243·10-s − 1.71i·11-s + (0.290 − 0.502i)12-s + (−0.879 − 0.235i)13-s + (0.856 + 0.856i)14-s + (0.386 − 0.103i)15-s + (0.124 − 0.216i)16-s + (0.226 + 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.32545 - 0.893656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.32545 - 0.893656i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.73 + 0.732i)T \) |
| 37 | \( 1 + (761. - 1.13e3i)T \) |
| good | 3 | \( 1 + (-9.05 + 5.22i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-8.32 - 2.22i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (-41.9 - 72.7i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + 207. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (148. + 39.8i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-65.3 - 243. i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (517. + 138. i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-278. + 278. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-567. - 567. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-347. - 347. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.86e3 - 1.07e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.40e3 + 1.40e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.17e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-1.87e3 + 3.24e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-962. - 3.59e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (78.0 - 291. i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-1.52e3 + 882. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (1.34e3 + 2.32e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + 8.29e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-7.42e3 - 1.99e3i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (3.89e3 - 6.74e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-3.64e3 + 977. i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-1.05e4 + 1.05e4i)T - 8.85e7iT^{2} \) |
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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
−13.80986893724554709470596827472, −12.81330940280948596359108603372, −11.80662630429616320939912548069, −10.59419945165534072727511173591, −8.696398206081553848868494163724, −8.256744467897296961898575254825, −6.36166559302629360925888723512, −5.14879252903369500006574084110, −2.99764500908927368026456981958, −1.99340302590035157606734953224,
2.12934863564230357315116085590, 4.01980094318414304846921006449, 4.79981574467403841761900269547, 7.01381221991727050048497004400, 7.904475769263897005333002836164, 9.541904023763642172142832043233, 10.37145004146774031603275938423, 11.87464247364024067274879328265, 13.18570745460832360776822932566, 14.19727614055715930236829352640
