| L(s) = 1 | + (−2.03 + 3.53i)2-s + (−10.2 − 17.7i)3-s + (7.67 + 13.3i)4-s + (36.0 + 62.4i)5-s + 83.5·6-s + (−123. − 213. i)7-s − 193.·8-s + (−88.2 + 152. i)9-s − 294.·10-s − 602.·11-s + (157. − 272. i)12-s + (−208. − 360. i)13-s + 1.00e3·14-s + (738. − 1.27e3i)15-s + (148. − 256. i)16-s + (−287. + 498. i)17-s + ⋯ |
| L(s) = 1 | + (−0.360 + 0.624i)2-s + (−0.656 − 1.13i)3-s + (0.239 + 0.415i)4-s + (0.645 + 1.11i)5-s + 0.947·6-s + (−0.950 − 1.64i)7-s − 1.06·8-s + (−0.363 + 0.629i)9-s − 0.930·10-s − 1.50·11-s + (0.315 − 0.546i)12-s + (−0.341 − 0.591i)13-s + 1.37·14-s + (0.847 − 1.46i)15-s + (0.144 − 0.250i)16-s + (−0.241 + 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0824564 - 0.224666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0824564 - 0.224666i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-7.34e3 - 3.92e3i)T \) |
| good | 2 | \( 1 + (2.03 - 3.53i)T + (-16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (10.2 + 17.7i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (-36.0 - 62.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (123. + 213. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + 602.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (208. + 360. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (287. - 498. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (393. + 681. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 - 1.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 360.T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-5.82e3 - 1.00e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + 7.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.75e4 + 3.03e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.92e4 - 3.32e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.87e4 + 3.24e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.08e4 - 1.87e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-7.96e3 - 1.38e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + 6.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.73e4 + 4.74e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.53e4 + 4.39e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-4.66e4 + 8.08e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 3.38e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.03639356125874876079208313134, −13.30543032381450948243607901157, −12.92206442892207062213088417563, −11.06295911554619000368940885831, −10.07913888749094786125338942940, −7.66159986071791572281650173271, −7.00720547866063035985913618924, −6.13350139192616855002111511690, −2.90071716181731084003697620367, −0.15166027036598514906613204385,
2.43323927123654750794453748726, 5.16380579552587022683241757439, 5.83385071483134422617659970445, 9.061171246994151287842108785685, 9.564248244466104666105030923504, 10.69900489957747505254103211161, 12.02881305995173272485471143216, 12.99473504562667528056773803820, 15.18301096013092895964494971664, 15.87018783353717495684779739656
