| L(s) = 1 | + (5.04 + 5.04i)2-s − 11.9i·3-s − 13.0i·4-s + (−47.3 + 47.3i)5-s + (60.2 − 60.2i)6-s + 421.·7-s + (388. − 388. i)8-s + 586.·9-s − 478.·10-s − 882. i·11-s − 155.·12-s + (1.09e3 − 1.09e3i)13-s + (2.12e3 + 2.12e3i)14-s + (564. + 564. i)15-s + 3.09e3·16-s + (−1.46e3 + 1.46e3i)17-s + ⋯ |
| L(s) = 1 | + (0.631 + 0.631i)2-s − 0.441i·3-s − 0.203i·4-s + (−0.378 + 0.378i)5-s + (0.278 − 0.278i)6-s + 1.22·7-s + (0.759 − 0.759i)8-s + 0.804·9-s − 0.478·10-s − 0.663i·11-s − 0.0898·12-s + (0.497 − 0.497i)13-s + (0.774 + 0.774i)14-s + (0.167 + 0.167i)15-s + 0.755·16-s + (−0.298 + 0.298i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.58414 - 0.140002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.58414 - 0.140002i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (4.19e4 - 2.83e4i)T \) |
| good | 2 | \( 1 + (-5.04 - 5.04i)T + 64iT^{2} \) |
| 3 | \( 1 + 11.9iT - 729T^{2} \) |
| 5 | \( 1 + (47.3 - 47.3i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 - 421.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 882. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + (-1.09e3 + 1.09e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (1.46e3 - 1.46e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + (2.18e3 - 2.18e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + (3.57e3 - 3.57e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + (2.43e4 + 2.43e4i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + (-2.25e4 - 2.25e4i)T + 8.87e8iT^{2} \) |
| 41 | \( 1 + 2.33e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (5.51e4 - 5.51e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.73e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 8.56e3T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-9.21e4 + 9.21e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-4.22e4 - 4.22e4i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 - 5.62e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.41e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.44e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + (5.22e5 - 5.22e5i)T - 2.43e11iT^{2} \) |
| 83 | \( 1 + 8.02e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-2.24e5 - 2.24e5i)T + 4.96e11iT^{2} \) |
| 97 | \( 1 + (-8.94e4 + 8.94e4i)T - 8.32e11iT^{2} \) |
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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
−15.07848826742708853122366308335, −13.99595489892673569425788508694, −13.05603659211257850167430356246, −11.45981969946553141845669666442, −10.32707092484344737525947046034, −8.201593025862058605048518852764, −7.07218255112765049733964732436, −5.65149334024221009490692401476, −4.11548135957308390618727824026, −1.37748378031351829478156263412,
1.85593288876605042494084251095, 4.08238760810887410690732841290, 4.82749656437557153758205417704, 7.42253510940935413431965286365, 8.745428974184459682550179743721, 10.54580550976383982761740254238, 11.59998573461240303149103563025, 12.57208278752554092961793057318, 13.78904133235850615088499468182, 15.00353976161874104469042808874
