| L(s) = 1 | + (−0.732 + 2.73i)2-s + (6.81 + 3.93i)3-s + (−6.92 − 4i)4-s + (−7.04 − 26.2i)5-s + (−15.7 + 15.7i)6-s + (41.4 − 71.8i)7-s + (16 − 15.9i)8-s + (−9.50 − 16.4i)9-s + 76.9·10-s − 62.6i·11-s + (−31.4 − 54.5i)12-s + (29.3 + 109. i)13-s + (165. + 165. i)14-s + (55.4 − 207. i)15-s + (31.9 + 55.4i)16-s + (−268. − 71.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (0.757 + 0.437i)3-s + (−0.433 − 0.250i)4-s + (−0.281 − 1.05i)5-s + (−0.437 + 0.437i)6-s + (0.846 − 1.46i)7-s + (0.250 − 0.249i)8-s + (−0.117 − 0.203i)9-s + 0.769·10-s − 0.517i·11-s + (−0.218 − 0.378i)12-s + (0.173 + 0.648i)13-s + (0.846 + 0.846i)14-s + (0.246 − 0.920i)15-s + (0.124 + 0.216i)16-s + (−0.928 − 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.68737 - 0.373364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68737 - 0.373364i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.732 - 2.73i)T \) |
| 37 | \( 1 + (279. - 1.34e3i)T \) |
| good | 3 | \( 1 + (-6.81 - 3.93i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (7.04 + 26.2i)T + (-541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (-41.4 + 71.8i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 62.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-29.3 - 109. i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (268. + 71.8i)T + (7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-122. - 456. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-545. + 545. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-835. - 835. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (644. + 644. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (436. + 252. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (758. - 758. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 171.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (913. + 1.58e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-4.48e3 - 1.20e3i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-3.63e3 + 975. i)T + (1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-4.64e3 - 2.68e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (2.04e3 - 3.53e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 - 5.79e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.42e3 - 5.32e3i)T + (-3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (4.42e3 + 7.65e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-3.42e3 + 1.27e4i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (8.92e3 - 8.92e3i)T - 8.85e7iT^{2} \) |
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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
−14.10023770947310468018533674524, −13.02880940394626490474448008362, −11.43712857654128489115682543081, −10.12103173063243353910885071688, −8.761117056985449769823403647420, −8.263644130283437334265371578021, −6.83380794722618963188850379200, −4.85587856299064453832058123508, −3.90292229585497083850885452566, −0.949591210718400052947741409569,
2.11256723795951421846201666812, 3.00654188887977061963498677598, 5.15916552561528520213639688205, 7.13537472453380890251492563697, 8.313004273148692875011659783523, 9.180402374508423287629816617876, 10.85709385691830710959273299826, 11.51597726870921248435321931567, 12.78593295726844673979622104018, 13.85109410030147231517926553973
