| L(s) = 1 | + (−1.05 + 3.94i)2-s + (0.0817 + 0.0471i)3-s + (−0.601 − 0.347i)4-s + (4.67 + 17.4i)5-s + (−0.272 + 0.272i)6-s + (−22.7 + 39.4i)7-s + (−44.2 + 44.2i)8-s + (−40.4 − 70.1i)9-s − 73.8·10-s − 53.3i·11-s + (−0.0327 − 0.0567i)12-s + (55.0 + 205. i)13-s + (−131. − 131. i)14-s + (−0.441 + 1.64i)15-s + (−133. − 230. i)16-s + (310. + 83.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.264 + 0.986i)2-s + (0.00907 + 0.00524i)3-s + (−0.0375 − 0.0216i)4-s + (0.187 + 0.698i)5-s + (−0.00757 + 0.00757i)6-s + (−0.465 + 0.805i)7-s + (−0.690 + 0.690i)8-s + (−0.499 − 0.865i)9-s − 0.738·10-s − 0.440i·11-s + (−0.000227 − 0.000393i)12-s + (0.325 + 1.21i)13-s + (−0.672 − 0.672i)14-s + (−0.00196 + 0.00731i)15-s + (−0.520 − 0.901i)16-s + (1.07 + 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.381063 + 1.15346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.381063 + 1.15346i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.35e3 - 200. i)T \) |
| good | 2 | \( 1 + (1.05 - 3.94i)T + (-13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-0.0817 - 0.0471i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-4.67 - 17.4i)T + (-541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (22.7 - 39.4i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 53.3iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-55.0 - 205. i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-310. - 83.0i)T + (7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-106. - 395. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-648. + 648. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (263. + 263. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (83.3 + 83.3i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-497. - 287. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.56e3 - 1.56e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 588.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.72e3 + 2.98e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (3.08e3 + 826. i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-3.62e3 + 970. i)T + (1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (1.48e3 + 855. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-1.47e3 + 2.55e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 - 5.90e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (88.3 + 329. i)T + (-3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (3.13e3 + 5.42e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.27e3 - 8.49e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-8.61e3 + 8.61e3i)T - 8.85e7iT^{2} \) |
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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
−16.22282981441512109371029696980, −14.84361489638594571588873029062, −14.38780839487761257597038337217, −12.39177209961749032249081230879, −11.31310178239615903990349043752, −9.474514811236421554478669511275, −8.348344304544340622370789792358, −6.66453046638476669914780965824, −5.92755253607100415237971084007, −3.02823986460636941939073802861,
0.964593280716215362867960408880, 3.10568078981367345187528879998, 5.34022648743843925028129808651, 7.39574436090034600720178362615, 9.161846393983152612810697744530, 10.31411260059215770021121137570, 11.27501308811902029031331730686, 12.72739595835302736758874511261, 13.49256208851047536343601165633, 15.25470573510363886435965295369
