| L(s) = 1 | + (2.35 + 3.23i)2-s + (−12.2 − 3.98i)3-s + (−4.94 + 15.2i)4-s + (55.8 − 1.65i)5-s + (−15.9 − 49.0i)6-s + 145. i·7-s + (−60.8 + 19.7i)8-s + (−62.1 − 45.1i)9-s + (136. + 176. i)10-s + (−424. + 308. i)11-s + (121. − 166. i)12-s + (−633. + 871. i)13-s + (−470. + 341. i)14-s + (−691. − 202. i)15-s + (−207. − 150. i)16-s + (1.38e3 − 451. i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.786 − 0.255i)3-s + (−0.154 + 0.475i)4-s + (0.999 − 0.0296i)5-s + (−0.180 − 0.556i)6-s + 1.12i·7-s + (−0.336 + 0.109i)8-s + (−0.255 − 0.185i)9-s + (0.432 + 0.559i)10-s + (−1.05 + 0.767i)11-s + (0.243 − 0.334i)12-s + (−1.03 + 1.42i)13-s + (−0.641 + 0.466i)14-s + (−0.793 − 0.232i)15-s + (−0.202 − 0.146i)16-s + (1.16 − 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.470362 + 1.18180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.470362 + 1.18180i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.35 - 3.23i)T \) |
| 5 | \( 1 + (-55.8 + 1.65i)T \) |
| good | 3 | \( 1 + (12.2 + 3.98i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 - 145. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (424. - 308. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (633. - 871. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.38e3 + 451. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-406. - 1.25e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.56e3 + 2.15e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (465. - 1.43e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.53e3 - 4.73e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-6.47e3 + 8.90e3i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.65e4 + 1.20e4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.62e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.47e4 - 4.80e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-3.05e4 - 9.92e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.55e3 + 2.58e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.99e3 + 1.45e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (3.23e4 - 1.05e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (4.22e3 - 1.30e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (9.19e3 + 1.26e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.27e4 - 7.01e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-9.30e4 + 3.02e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (5.79e4 - 4.20e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-7.01e4 - 2.27e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.81166867070604055934926465968, −14.03118123174130029575836426761, −12.36055718486042129819054745248, −12.11891664118467682564100174982, −10.16330409665294558450891286999, −8.933918973130905361323981957440, −7.14849790319861540990388139503, −5.85011012016251543530984753014, −5.09302125218242534095378129716, −2.34893047463255880448488080800,
0.60609089492103871045848753868, 2.91466722994939451357085166644, 5.05153999939773589113290456949, 5.85493141799587366425839957284, 7.82475890599483275771206942327, 10.11389131865662593410634285414, 10.35373499454895006750040182738, 11.65213142121145101248835900699, 13.16022966448481637455550243653, 13.71689369850384686070162273834
