L(s) = 1 | + (−2 − 2i)2-s − 5.34i·3-s + 8i·4-s + (10.1 − 10.1i)5-s + (−10.6 + 10.6i)6-s − 94.0·7-s + (16 − 16i)8-s + 52.4·9-s − 40.7·10-s − 104. i·11-s + 42.7·12-s + (−156. + 156. i)13-s + (188. + 188. i)14-s + (−54.4 − 54.4i)15-s − 64·16-s + (−288. + 288. i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s − 0.593i·3-s + 0.5i·4-s + (0.407 − 0.407i)5-s + (−0.296 + 0.296i)6-s − 1.92·7-s + (0.250 − 0.250i)8-s + 0.647·9-s − 0.407·10-s − 0.865i·11-s + 0.296·12-s + (−0.928 + 0.928i)13-s + (0.960 + 0.960i)14-s + (−0.241 − 0.241i)15-s − 0.250·16-s + (−0.998 + 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0764215 + 0.192023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0764215 + 0.192023i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 + 2i)T \) |
| 37 | \( 1 + (913. + 1.01e3i)T \) |
good | 3 | \( 1 + 5.34iT - 81T^{2} \) |
| 5 | \( 1 + (-10.1 + 10.1i)T - 625iT^{2} \) |
| 7 | \( 1 + 94.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + 104. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (156. - 156. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (288. - 288. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + (257. - 257. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + (-226. + 226. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (59.2 + 59.2i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (608. + 608. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + 1.81e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.88e3 + 1.88e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.51e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 185.T + 7.89e6T^{2} \) |
| 59 | \( 1 + (1.98e3 - 1.98e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 + (4.80e3 + 4.80e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + 2.62e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.72e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 7.87e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-3.21e3 + 3.21e3i)T - 3.89e7iT^{2} \) |
| 83 | \( 1 + 9.87e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-3.65e3 - 3.65e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (-945. + 945. i)T - 8.85e7iT^{2} \) |
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
−12.80153914295967562438755122347, −12.46731075110566770426760302143, −10.74944302718198205043127720292, −9.646385439106593560833623169438, −8.848047337361044631904234483748, −7.12844157435768736294782743162, −6.17676702272708505280835142726, −3.84942072193650612333145556537, −2.06099650173271153187448753641, −0.11703406399883178732861041224,
2.82541742880188423561467186368, 4.77377963994803601405298060485, 6.48875639461906434805582356716, 7.21821240842939001821877723241, 9.284251286989555447111480406905, 9.802015921557124814342972162483, 10.62540770039526930089465118916, 12.57138741270628830747664221390, 13.36140408992317192074290145917, 14.93709066167101975934656503969