| L(s) = 1 | + (5.01 + 28.4i)3-s + (−31.8 + 26.6i)5-s + (−9.54 + 16.5i)7-s + (−555. + 202. i)9-s + (368. + 638. i)11-s + (183. − 1.04e3i)13-s + (−918. − 770. i)15-s + (−1.01e3 − 369. i)17-s + (−458. − 1.50e3i)19-s + (−518. − 188. i)21-s + (2.27e3 + 1.90e3i)23-s + (−243. + 1.38e3i)25-s + (−5.02e3 − 8.70e3i)27-s + (−3.17e3 + 1.15e3i)29-s + (−1.28e3 + 2.22e3i)31-s + ⋯ |
| L(s) = 1 | + (0.321 + 1.82i)3-s + (−0.568 + 0.477i)5-s + (−0.0736 + 0.127i)7-s + (−2.28 + 0.831i)9-s + (0.918 + 1.59i)11-s + (0.301 − 1.70i)13-s + (−1.05 − 0.884i)15-s + (−0.853 − 0.310i)17-s + (−0.291 − 0.956i)19-s + (−0.256 − 0.0933i)21-s + (0.897 + 0.752i)23-s + (−0.0778 + 0.441i)25-s + (−1.32 − 2.29i)27-s + (−0.700 + 0.254i)29-s + (−0.240 + 0.416i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.159444 - 1.23552i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159444 - 1.23552i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (458. + 1.50e3i)T \) |
| good | 3 | \( 1 + (-5.01 - 28.4i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (31.8 - 26.6i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (9.54 - 16.5i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-368. - 638. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-183. + 1.04e3i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (1.01e3 + 369. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 23 | \( 1 + (-2.27e3 - 1.90e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (3.17e3 - 1.15e3i)T + (1.57e7 - 1.31e7i)T^{2} \) |
| 31 | \( 1 + (1.28e3 - 2.22e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 163.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (531. + 3.01e3i)T + (-1.08e8 + 3.96e7i)T^{2} \) |
| 43 | \( 1 + (1.24e4 - 1.04e4i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-473. + 172. i)T + (1.75e8 - 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-1.11e4 - 9.31e3i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-2.42e4 - 8.83e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-2.43e4 - 2.04e4i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (1.57e4 - 5.72e3i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (1.54e4 - 1.29e4i)T + (3.13e8 - 1.77e9i)T^{2} \) |
| 73 | \( 1 + (-1.13e3 - 6.45e3i)T + (-1.94e9 + 7.09e8i)T^{2} \) |
| 79 | \( 1 + (3.95e3 + 2.24e4i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (6.10e4 - 1.05e5i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-9.89e3 + 5.61e4i)T + (-5.24e9 - 1.90e9i)T^{2} \) |
| 97 | \( 1 + (3.68e4 + 1.34e4i)T + (6.57e9 + 5.51e9i)T^{2} \) |
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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
−14.73885110519452658458861758131, −13.10883617011306364416281738740, −11.51097216439473268158524984737, −10.69419544581502154934787012382, −9.666994951495250583427235562983, −8.788835392700313507363011002355, −7.22103502189350321118061870593, −5.25306493859418604718645222709, −4.11990340311799852556464458784, −2.96228662973179677053993610868,
0.51893473205203792954313362133, 1.86039611030621899909355578468, 3.79652650556847433590894756792, 6.16241243165623352045900323939, 6.92401883860525332128319459906, 8.409410566255054883658750267174, 8.829649846522155115782152698728, 11.33019101572168247720475609666, 11.87268064163226539913012003261, 13.02857138681993948552504725181
