| L(s) = 1 | + (−6.59 − 6.59i)2-s + 40.5i·3-s + 22.9i·4-s + (83.9 − 83.9i)5-s + (267. − 267. i)6-s − 26.0·7-s + (−270. + 270. i)8-s − 912.·9-s − 1.10e3·10-s + 1.07e3i·11-s − 929.·12-s + (−521. + 521. i)13-s + (171. + 171. i)14-s + (3.39e3 + 3.39e3i)15-s + 5.03e3·16-s + (−3.41e3 + 3.41e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 − 0.824i)2-s + 1.50i·3-s + 0.358i·4-s + (0.671 − 0.671i)5-s + (1.23 − 1.23i)6-s − 0.0758·7-s + (−0.528 + 0.528i)8-s − 1.25·9-s − 1.10·10-s + 0.809i·11-s − 0.538·12-s + (−0.237 + 0.237i)13-s + (0.0624 + 0.0624i)14-s + (1.00 + 1.00i)15-s + 1.23·16-s + (−0.695 + 0.695i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.422522 + 0.537265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422522 + 0.537265i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-4.09e4 - 2.98e4i)T \) |
| good | 2 | \( 1 + (6.59 + 6.59i)T + 64iT^{2} \) |
| 3 | \( 1 - 40.5iT - 729T^{2} \) |
| 5 | \( 1 + (-83.9 + 83.9i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + 26.0T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.07e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + (521. - 521. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (3.41e3 - 3.41e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + (7.19e3 - 7.19e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + (5.40e3 - 5.40e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + (2.70e4 + 2.70e4i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + (-1.19e4 - 1.19e4i)T + 8.87e8iT^{2} \) |
| 41 | \( 1 + 2.28e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-7.97e4 + 7.97e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 2.63e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.96e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.42e5 + 1.42e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-7.53e4 - 7.53e4i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 - 4.41e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.99e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 5.23e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + (-8.97e3 + 8.97e3i)T - 2.43e11iT^{2} \) |
| 83 | \( 1 + 2.36e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (7.63e5 + 7.63e5i)T + 4.96e11iT^{2} \) |
| 97 | \( 1 + (-1.01e6 + 1.01e6i)T - 8.32e11iT^{2} \) |
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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
−15.49592480105933357901678086397, −14.55768687732852008220277949364, −12.76153011087225423188962460356, −11.29556731852646453665462750641, −10.10751052250323015960363519240, −9.617689101540551150630001306731, −8.542764894159418580823542983759, −5.68643638776530097373031902973, −4.17289289251036311667474726452, −1.95964097818664041290402640974,
0.42730202761324515218875012139, 2.52575688638461682904627139653, 6.15950674661472179137967967256, 6.86599482428581677925171878543, 8.009116256352619159088358290332, 9.252242213780035181506735851676, 11.01390919916543784367415113489, 12.61071372670954896307030152961, 13.54581835312186542071778376261, 14.75151556656654935152766070994
