| L(s) = 1 | + (3.59 − 0.633i)2-s + (−1.19 + 6.77i)3-s + (5.00 − 1.82i)4-s + (7.95 − 9.48i)5-s + 25.0i·6-s + (3.25 + 2.72i)7-s + (−8.46 + 4.88i)8-s + (−19.0 − 6.93i)9-s + (22.5 − 39.1i)10-s + (−17.6 − 30.5i)11-s + (6.35 + 36.0i)12-s + (−27.9 − 76.8i)13-s + (13.4 + 7.74i)14-s + (54.7 + 65.1i)15-s + (−59.9 + 50.3i)16-s + (−24.0 + 65.9i)17-s + ⋯ |
| L(s) = 1 | + (1.27 − 0.224i)2-s + (−0.229 + 1.30i)3-s + (0.625 − 0.227i)4-s + (0.711 − 0.848i)5-s + 1.70i·6-s + (0.175 + 0.147i)7-s + (−0.374 + 0.216i)8-s + (−0.705 − 0.256i)9-s + (0.714 − 1.23i)10-s + (−0.483 − 0.837i)11-s + (0.152 + 0.866i)12-s + (−0.596 − 1.63i)13-s + (0.256 + 0.147i)14-s + (0.941 + 1.12i)15-s + (−0.936 + 0.785i)16-s + (−0.342 + 0.941i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.09766 + 0.460595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09766 + 0.460595i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-38.6 + 221. i)T \) |
| good | 2 | \( 1 + (-3.59 + 0.633i)T + (7.51 - 2.73i)T^{2} \) |
| 3 | \( 1 + (1.19 - 6.77i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (-7.95 + 9.48i)T + (-21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-3.25 - 2.72i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (17.6 + 30.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (27.9 + 76.8i)T + (-1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (24.0 - 65.9i)T + (-3.76e3 - 3.15e3i)T^{2} \) |
| 19 | \( 1 + (-71.4 - 12.6i)T + (6.44e3 + 2.34e3i)T^{2} \) |
| 23 | \( 1 + (-111. - 64.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-27.5 + 15.9i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 105. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (463. - 168. i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 - 71.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (6.20 - 10.7i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-229. + 192. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (375. + 447. i)T + (-3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-276. - 758. i)T + (-1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-752. - 631. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-128. + 726. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 - 905.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (139. - 166. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (381. + 138. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (193. + 230. i)T + (-1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (872. + 503. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.63268797549401067601730800270, −14.87405079725821106043921197085, −13.44938951085350196990191100274, −12.68774319892026516600042478984, −11.15746726738650925416066193206, −9.966400496532722893288431779803, −8.592174881358549763024180203668, −5.50499406752081755769567531083, −5.10387717914586549541075828422, −3.35415576147220995350148955535,
2.40613794419447715696280497386, 4.90537546949340932536913618530, 6.61163926109307814570217106090, 7.11125666359275268534036357143, 9.615389165097036843494572267429, 11.53367672789634972277401045095, 12.48679586100549285672167749822, 13.68315161665366143645988563109, 14.09568011872650583393648569194, 15.35128341329086002167709916346
