| L(s) = 1 | + (2.73 + 0.732i)2-s + (−15.3 − 8.88i)3-s + (6.92 + 4i)4-s + (−1.55 + 0.417i)5-s + (−35.5 − 35.5i)6-s + (−4.44 + 7.69i)7-s + (15.9 + 16i)8-s + (117. + 203. i)9-s − 4.55·10-s + 118. i·11-s + (−71.0 − 123. i)12-s + (−175. + 46.8i)13-s + (−17.7 + 17.7i)14-s + (27.6 + 7.41i)15-s + (31.9 + 55.4i)16-s + (−93.1 + 347. i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (−1.71 − 0.987i)3-s + (0.433 + 0.250i)4-s + (−0.0622 + 0.0166i)5-s + (−0.987 − 0.987i)6-s + (−0.0906 + 0.156i)7-s + (0.249 + 0.250i)8-s + (1.45 + 2.51i)9-s − 0.0455·10-s + 0.976i·11-s + (−0.493 − 0.855i)12-s + (−1.03 + 0.277i)13-s + (−0.0906 + 0.0906i)14-s + (0.122 + 0.0329i)15-s + (0.124 + 0.216i)16-s + (−0.322 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.526273 + 0.600314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.526273 + 0.600314i\) |
| \(L(3)\) |
|
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 | \( 1 + (1.34e3 - 266. i)T \) |
| good | 3 | \( 1 + (15.3 + 8.88i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (1.55 - 0.417i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (4.44 - 7.69i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 - 118. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (175. - 46.8i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (93.1 - 347. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-476. + 127. i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (194. + 194. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (74.3 - 74.3i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (352. - 352. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-41.3 - 23.8i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-311. - 311. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 3.38e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-2.63e3 - 4.56e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-1.29e3 + 4.83e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (116. + 436. i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (1.32e3 + 765. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (878. - 1.52e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 934. iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (4.51e3 - 1.20e3i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-2.01e3 - 3.48e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.01e4 - 2.72e3i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (4.49e3 + 4.49e3i)T + 8.85e7iT^{2} \) |
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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
−13.81603547996002360415502813856, −12.68048693873175148713177434885, −12.18327174969602362725934307303, −11.25560035160757678576982247447, −10.04188473043587752330317011237, −7.62442658179891790690476821217, −6.84027981477432368543257029466, −5.67041618442525591233635749548, −4.64031186870029777441087127939, −1.82845033019910750520056342731,
0.40231568283711950632326148366, 3.61280442034721227556620944844, 5.00418866123946802166831131525, 5.76433981521097916867033010818, 7.12612701473685670308308292894, 9.575349482452929772171623440954, 10.38495676880192978089744331883, 11.66157022640979121509829971465, 11.87915904051090897319430372496, 13.40870880237670832775190152917
