| L(s) = 1 | + (−1.09 − 0.193i)2-s + (0.594 + 3.37i)3-s + (−6.35 − 2.31i)4-s + (12.8 + 15.3i)5-s − 3.80i·6-s + (−19.5 + 16.4i)7-s + (14.2 + 8.20i)8-s + (14.3 − 5.22i)9-s + (−11.1 − 19.2i)10-s + (−11.6 + 20.1i)11-s + (4.02 − 22.8i)12-s + (14.4 − 39.6i)13-s + (24.5 − 14.1i)14-s + (−43.9 + 52.4i)15-s + (27.4 + 23.0i)16-s + (−34.2 − 94.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.387 − 0.0682i)2-s + (0.114 + 0.648i)3-s + (−0.794 − 0.289i)4-s + (1.14 + 1.37i)5-s − 0.258i·6-s + (−1.05 + 0.886i)7-s + (0.628 + 0.362i)8-s + (0.531 − 0.193i)9-s + (−0.351 − 0.608i)10-s + (−0.319 + 0.553i)11-s + (0.0967 − 0.548i)12-s + (0.307 − 0.845i)13-s + (0.469 − 0.271i)14-s + (−0.757 + 0.902i)15-s + (0.429 + 0.360i)16-s + (−0.488 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.744580 + 0.662814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744580 + 0.662814i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-198. + 105. i)T \) |
| good | 2 | \( 1 + (1.09 + 0.193i)T + (7.51 + 2.73i)T^{2} \) |
| 3 | \( 1 + (-0.594 - 3.37i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (-12.8 - 15.3i)T + (-21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (19.5 - 16.4i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (11.6 - 20.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-14.4 + 39.6i)T + (-1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (34.2 + 94.0i)T + (-3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (-44.0 + 7.77i)T + (6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (-51.4 + 29.7i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-164. - 95.0i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 126. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (82.3 + 29.9i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + 376. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-86.1 - 149. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (359. + 301. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-324. + 387. i)T + (-3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (65.9 - 181. i)T + (-1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (481. - 404. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (15.8 + 90.0i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 - 44.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-319. - 380. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-481. + 175. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (278. - 331. i)T + (-1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (639. - 369. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.96665466013834983855086438567, −15.03163782272437161928792389820, −13.89972430297869697907897673868, −12.82443594310541930579669244410, −10.62989144866831711431770687158, −9.854008932407643914794592293484, −9.178355989914984245986776427917, −6.84393491125654229636469275002, −5.31197173920931586777733530548, −2.92957294192932161748447435716,
1.12161775745292223194905099043, 4.38702892852995503961417796290, 6.32111095121413899252138392382, 8.066935336663701772453458794398, 9.272423634113321800906628285465, 10.16034226130656664061901175762, 12.60190907403109896673138926045, 13.43170638504183349803147011498, 13.59159881178960298547255817549, 16.23678473323026027185402844918
