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
−13.40870880237670832775190152917, −11.87915904051090897319430372496, −11.66157022640979121509829971465, −10.38495676880192978089744331883, −9.575349482452929772171623440954, −7.12612701473685670308308292894, −5.76433981521097916867033010818, −5.00418866123946802166831131525, −3.61280442034721227556620944844, −0.40231568283711950632326148366,
1.82845033019910750520056342731, 4.64031186870029777441087127939, 5.67041618442525591233635749548, 6.84027981477432368543257029466, 7.62442658179891790690476821217, 10.04188473043587752330317011237, 11.25560035160757678576982247447, 12.18327174969602362725934307303, 12.68048693873175148713177434885, 13.81603547996002360415502813856