L(s) = 1 | + 4.18i·2-s + 63.9·3-s + 110.·4-s + 270. i·5-s + 267. i·6-s + 565.·7-s + 998. i·8-s + 1.90e3·9-s − 1.13e3·10-s − 4.24e3·11-s + 7.06e3·12-s − 1.35e4i·13-s + 2.36e3i·14-s + 1.72e4i·15-s + 9.96e3·16-s + 1.40e4i·17-s + ⋯ |
L(s) = 1 | + 0.370i·2-s + 1.36·3-s + 0.863·4-s + 0.967i·5-s + 0.506i·6-s + 0.623·7-s + 0.689i·8-s + 0.870·9-s − 0.357·10-s − 0.961·11-s + 1.18·12-s − 1.71i·13-s + 0.230i·14-s + 1.32i·15-s + 0.608·16-s + 0.695i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.93020 + 1.51349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.93020 + 1.51349i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-1.78e5 + 2.51e5i)T \) |
good | 2 | \( 1 - 4.18iT - 128T^{2} \) |
| 3 | \( 1 - 63.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 270. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 565.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.24e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.35e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.40e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 2.43e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.87e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.59e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.56e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + 3.75e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.26e3iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 4.51e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.43e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.76e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.41e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 3.53e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.72e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.22e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.18e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 6.56e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.16e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 7.56e6iT - 8.07e13T^{2} \) |
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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
−15.05134772851286310458003998636, −14.27992892649053777689662951715, −12.90867756045998796246740038684, −11.10700644124888780494548454938, −10.13557726027437668305921771296, −8.014528383055091904209902161978, −7.72962696530032172444796661528, −5.84485170073968124157825635459, −3.24281598618540462347545610688, −2.22357189971039424985398751398,
1.56059226140633776907760191077, 2.81898674213608796271804244912, 4.70203318386728771625796670730, 7.09174961768016664915210411534, 8.408774250036982369492144715217, 9.390736890313983441717252435357, 11.04270988891925338496504614583, 12.30626103885169041473289615224, 13.53004290512118989562779893772, 14.58854795610156348409930232031