| L(s) = 1 | + (1.50 + 4.12i)2-s + (2.69 + 0.982i)3-s + (−8.63 + 7.24i)4-s + (2.23 − 0.394i)5-s + 12.6i·6-s + (−1.77 − 10.0i)7-s + (−12.4 − 7.19i)8-s + (−14.3 − 12.0i)9-s + (4.98 + 8.63i)10-s + (7.09 − 12.2i)11-s + (−30.4 + 11.0i)12-s + (24.4 + 29.1i)13-s + (38.9 − 22.4i)14-s + (6.42 + 1.13i)15-s + (−4.69 + 26.6i)16-s + (50.9 − 60.6i)17-s + ⋯ |
| L(s) = 1 | + (0.530 + 1.45i)2-s + (0.519 + 0.189i)3-s + (−1.07 + 0.906i)4-s + (0.199 − 0.0352i)5-s + 0.858i·6-s + (−0.0959 − 0.544i)7-s + (−0.550 − 0.317i)8-s + (−0.531 − 0.446i)9-s + (0.157 + 0.272i)10-s + (0.194 − 0.337i)11-s + (−0.732 + 0.266i)12-s + (0.521 + 0.621i)13-s + (0.742 − 0.428i)14-s + (0.110 + 0.0194i)15-s + (−0.0733 + 0.416i)16-s + (0.726 − 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.10275 + 1.43248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10275 + 1.43248i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-158. + 159. i)T \) |
| good | 2 | \( 1 + (-1.50 - 4.12i)T + (-6.12 + 5.14i)T^{2} \) |
| 3 | \( 1 + (-2.69 - 0.982i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-2.23 + 0.394i)T + (117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (1.77 + 10.0i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-7.09 + 12.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-24.4 - 29.1i)T + (-381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-50.9 + 60.6i)T + (-853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-10.5 + 28.9i)T + (-5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (32.1 - 18.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (169. + 98.1i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 193. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (342. - 287. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 - 56.2iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (120. + 209. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (0.249 - 1.41i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-571. - 100. i)T + (1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-14.7 - 17.5i)T + (-3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-157. - 891. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-711. - 258. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + 225.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-1.01e3 + 178. i)T + (4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-350. - 294. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (1.18e3 + 208. i)T + (6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (1.13e3 - 655. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−16.14214797278543020664196063337, −14.98897587857583569608741950850, −14.05790237723998447987180355696, −13.42299979489038550338708896252, −11.52799735029430862813376214471, −9.540550917779342244212377408286, −8.280307860318636471871441261096, −6.93991642526691642141992045867, −5.60640467406023912613701861306, −3.80355287603276027509694064212,
2.03999264368121388874820666551, 3.57217170763069336669399749734, 5.60145641749005441156691199780, 8.073452990367937954728902946595, 9.600233156018590889938670856084, 10.79775907556021614352885978693, 12.00535618425366508789618902364, 13.02102758256494274875226410806, 13.97050198182960969882112590763, 15.09595349748373262601062714338
