L(s) = 1 | + (1.61 + 1.86i)2-s + (1.97 + 1.26i)3-s + (0.275 − 1.91i)4-s + (−3.07 + 6.73i)5-s + (0.822 + 5.71i)6-s + (−11.6 − 3.43i)7-s + (20.5 − 13.2i)8-s + (−8.92 − 19.5i)9-s + (−17.4 + 5.13i)10-s + (3.34 − 3.85i)11-s + (2.97 − 3.43i)12-s + (−23.2 + 6.83i)13-s + (−12.4 − 27.2i)14-s + (−14.6 + 9.39i)15-s + (42.9 + 12.6i)16-s + (9.15 + 63.6i)17-s + ⋯ |
L(s) = 1 | + (0.570 + 0.657i)2-s + (0.379 + 0.244i)3-s + (0.0344 − 0.239i)4-s + (−0.275 + 0.602i)5-s + (0.0559 + 0.389i)6-s + (−0.631 − 0.185i)7-s + (0.909 − 0.584i)8-s + (−0.330 − 0.724i)9-s + (−0.553 + 0.162i)10-s + (0.0916 − 0.105i)11-s + (0.0716 − 0.0826i)12-s + (−0.496 + 0.145i)13-s + (−0.237 − 0.521i)14-s + (−0.251 + 0.161i)15-s + (0.670 + 0.197i)16-s + (0.130 + 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.39952 + 0.533774i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39952 + 0.533774i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (-72.4 - 83.1i)T \) |
good | 2 | \( 1 + (-1.61 - 1.86i)T + (-1.13 + 7.91i)T^{2} \) |
| 3 | \( 1 + (-1.97 - 1.26i)T + (11.2 + 24.5i)T^{2} \) |
| 5 | \( 1 + (3.07 - 6.73i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (11.6 + 3.43i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (-3.34 + 3.85i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (23.2 - 6.83i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-9.15 - 63.6i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (7.05 - 49.0i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (24.5 + 170. i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (195. - 125. i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (98.8 + 216. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-129. + 283. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-442. - 284. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 - 407.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (461. + 135. i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-41.1 + 12.0i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (286. - 183. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (347. + 400. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-488. - 563. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-43.0 + 299. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (744. - 218. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-121. - 266. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-479. - 307. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-776. + 1.70e3i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.24604963376440013166099757606, −15.87983503760406597634044247633, −14.90652784503063679155362476609, −14.18230859175537743267135965748, −12.66016004852257040562165520870, −10.79402099183297901768161297915, −9.407029431723990040804514585269, −7.30371093670489269138934580083, −5.95505189693845551092908837608, −3.72785995946286649433956837095,
2.78500731338958596124620794463, 4.84370593803020163814755552561, 7.46492861007565883041476240731, 8.953509415801991109887457010653, 10.92109110003083028746170990696, 12.32855101348388968192382960198, 13.10612674973092973793398111796, 14.31943540474803419145332088585, 16.11704702488229372262985398107, 17.02806826603081328108874359777