| L(s) = 1 | + (11.2 + 11.2i)2-s − 11.8i·3-s − 1.30i·4-s + (−14.3 + 14.3i)5-s + (133. − 133. i)6-s − 997.·7-s + (2.90e3 − 2.90e3i)8-s + 6.42e3·9-s − 323.·10-s + 862. i·11-s − 15.5·12-s + (3.22e4 − 3.22e4i)13-s + (−1.12e4 − 1.12e4i)14-s + (170. + 170. i)15-s + 6.51e4·16-s + (3.56e4 − 3.56e4i)17-s + ⋯ |
| L(s) = 1 | + (0.705 + 0.705i)2-s − 0.146i·3-s − 0.00511i·4-s + (−0.0229 + 0.0229i)5-s + (0.103 − 0.103i)6-s − 0.415·7-s + (0.708 − 0.708i)8-s + 0.978·9-s − 0.0323·10-s + 0.0589i·11-s − 0.000747·12-s + (1.12 − 1.12i)13-s + (−0.292 − 0.292i)14-s + (0.00335 + 0.00335i)15-s + 0.994·16-s + (0.427 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0819i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.996 + 0.0819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.86973 - 0.117724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.86973 - 0.117724i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-1.52e6 + 1.09e6i)T \) |
| good | 2 | \( 1 + (-11.2 - 11.2i)T + 256iT^{2} \) |
| 3 | \( 1 + 11.8iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (14.3 - 14.3i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + 997.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 862. iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (-3.22e4 + 3.22e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (-3.56e4 + 3.56e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (-8.32e3 + 8.32e3i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + (-8.32e3 + 8.32e3i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (-7.34e5 - 7.34e5i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + (1.06e6 + 1.06e6i)T + 8.52e11iT^{2} \) |
| 41 | \( 1 + 2.78e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (3.38e6 - 3.38e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + 2.64e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 7.81e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (3.14e6 - 3.14e6i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (-9.02e6 - 9.02e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 - 5.95e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 4.67e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 8.36e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (2.63e7 - 2.63e7i)T - 1.51e15iT^{2} \) |
| 83 | \( 1 - 2.22e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-4.19e7 - 4.19e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (9.66e7 - 9.66e7i)T - 7.83e15iT^{2} \) |
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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
−14.69613885755158987002239848969, −13.35250246996888539410287453441, −12.76078871398031374102878992889, −10.83773297326316601949439495841, −9.638603139552817523251763440098, −7.71446158608334531132778102545, −6.51473677993818469854355399210, −5.24807426477056603238241964416, −3.63440866589841438732249739728, −1.10204500152341642724006112949,
1.63053803134424833422257935395, 3.46606118778000147706467931063, 4.57109370428267814844035127417, 6.54288494017001229728569419883, 8.261560389236877118097529554600, 9.910931648111734532171856493245, 11.16617686432004451774277392943, 12.31408422074230824750025094180, 13.26227261907302795111184814796, 14.24703435999036850602322876943
