| L(s) = 1 | + (−1.35 − 5.06i)2-s + (−10.5 + 6.10i)3-s + (−9.99 + 5.77i)4-s + (0.491 − 1.83i)5-s + (45.3 + 45.3i)6-s + (44.1 + 76.4i)7-s + (−16.5 − 16.5i)8-s + (34.0 − 59.0i)9-s − 9.97·10-s + 127. i·11-s + (70.4 − 122. i)12-s + (7.06 − 26.3i)13-s + (327. − 327. i)14-s + (6.00 + 22.4i)15-s + (−153. + 266. i)16-s + (−283. + 75.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.339 − 1.26i)2-s + (−1.17 + 0.678i)3-s + (−0.624 + 0.360i)4-s + (0.0196 − 0.0734i)5-s + (1.25 + 1.25i)6-s + (0.900 + 1.55i)7-s + (−0.258 − 0.258i)8-s + (0.420 − 0.729i)9-s − 0.0997·10-s + 1.05i·11-s + (0.489 − 0.847i)12-s + (0.0417 − 0.155i)13-s + (1.67 − 1.67i)14-s + (0.0266 + 0.0996i)15-s + (−0.600 + 1.04i)16-s + (−0.981 + 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.586826 + 0.261507i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586826 + 0.261507i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.28e3 - 465. i)T \) |
| good | 2 | \( 1 + (1.35 + 5.06i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (10.5 - 6.10i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-0.491 + 1.83i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-44.1 - 76.4i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 127. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-7.06 + 26.3i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (283. - 75.9i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (131. - 491. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (262. + 262. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-37.7 + 37.7i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-674. + 674. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-980. + 566. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (223. + 223. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 262.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.78e3 - 3.08e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-6.07e3 + 1.62e3i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (1.14e3 + 307. i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (4.73e3 - 2.73e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-2.81e3 - 4.88e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + 1.02e4iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.06e3 - 3.99e3i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-978. + 1.69e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.70e3 + 1.00e4i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-9.90e3 - 9.90e3i)T + 8.85e7iT^{2} \) |
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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
−15.75190681269546162190217763349, −14.91521398261964948439681718884, −12.52945118835655453997488855279, −11.92019655467156842148260656917, −10.97062448926076223929342834993, −9.984134964127690549108378216605, −8.654348588446316951790912394228, −6.00147328904504179473357652496, −4.57136207886810906383076927328, −2.05515698743712618463529180332,
0.56652524004842072433651007570, 4.92281791713064704753847940693, 6.47881418888510273155502807282, 7.20719920566491055655136429216, 8.552092967246485943571681209296, 10.82641392011606495199839026776, 11.54319771728525792331499491565, 13.38845591425387207697766173883, 14.32384249796897256480031530668, 15.90342899525153318378352560618
