| L(s) = 1 | + (3.23 + 2.35i)2-s + (1.57 + 4.86i)3-s + (4.94 + 15.2i)4-s + (−52.9 + 18.0i)5-s + (−6.31 + 19.4i)6-s − 251.·7-s + (−19.7 + 60.8i)8-s + (175. − 127. i)9-s + (−213. − 66.1i)10-s + (261. + 190. i)11-s + (−66.1 + 48.0i)12-s + (−631. + 459. i)13-s + (−814. − 591. i)14-s + (−171. − 228. i)15-s + (−207. + 150. i)16-s + (190. − 587. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.101 + 0.311i)3-s + (0.154 + 0.475i)4-s + (−0.946 + 0.322i)5-s + (−0.0716 + 0.220i)6-s − 1.94·7-s + (−0.109 + 0.336i)8-s + (0.722 − 0.524i)9-s + (−0.675 − 0.209i)10-s + (0.652 + 0.473i)11-s + (−0.132 + 0.0963i)12-s + (−1.03 + 0.753i)13-s + (−1.11 − 0.807i)14-s + (−0.196 − 0.262i)15-s + (−0.202 + 0.146i)16-s + (0.160 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0978i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0464143 + 0.946279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0464143 + 0.946279i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 5 | \( 1 + (52.9 - 18.0i)T \) |
| good | 3 | \( 1 + (-1.57 - 4.86i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + 251.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-261. - 190. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (631. - 459. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-190. + 587. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (623. - 1.91e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (9.99e2 + 726. i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-2.62e3 - 8.08e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-63.5 + 195. i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.11e3 - 811. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (708. - 514. i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.49e3 + 7.67e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (3.85e3 + 1.18e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.31e4 - 2.40e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (4.43e4 + 3.21e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.36e4 - 4.21e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (4.65e3 + 1.43e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.22e4 + 8.87e3i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.51e4 - 4.64e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (1.67e4 - 5.16e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-3.57e4 - 2.60e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (3.34e4 + 1.02e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−15.14370402130931852759414468134, −14.21188952286071166130557615827, −12.45527131527146317046911872922, −12.26114716222249190895683617758, −10.22159605771869181874590547413, −9.189912917506503383345886829890, −7.21106627280393370524134252581, −6.51505755373486307099851850926, −4.29420560727379257673783910508, −3.26504499070974690577184260673,
0.38682991216756523998521827221, 2.93969998602576346362751291579, 4.32629237958598204496002990226, 6.27686315569857395304710768784, 7.54494672493429985004527818292, 9.334808150441537476402869877324, 10.49719099336150845700996772665, 12.06976768597207809682366165800, 12.77533261692208476998169461295, 13.59584144381767220205623597664
