| L(s) = 1 | + (0.694 + 3.93i)2-s + (−21.8 + 18.3i)3-s + (−15.0 + 5.47i)4-s + (33.2 + 12.0i)5-s + (−87.3 − 73.2i)6-s + (−42.9 + 74.3i)7-s + (−32 − 55.4i)8-s + (98.8 − 560. i)9-s + (−24.5 + 139. i)10-s + (−299. − 519. i)11-s + (228. − 394. i)12-s + (417. + 350. i)13-s + (−322. − 117. i)14-s + (−946. + 344. i)15-s + (196. − 164. i)16-s + (252. + 1.43e3i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−1.40 + 1.17i)3-s + (−0.469 + 0.171i)4-s + (0.594 + 0.216i)5-s + (−0.990 − 0.831i)6-s + (−0.331 + 0.573i)7-s + (−0.176 − 0.306i)8-s + (0.406 − 2.30i)9-s + (−0.0776 + 0.440i)10-s + (−0.747 − 1.29i)11-s + (0.457 − 0.791i)12-s + (0.685 + 0.574i)13-s + (−0.440 − 0.160i)14-s + (−1.08 + 0.395i)15-s + (0.191 − 0.160i)16-s + (0.212 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.220815 - 0.351081i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.220815 - 0.351081i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.694 - 3.93i)T \) |
| 19 | \( 1 + (1.57e3 - 94.1i)T \) |
| good | 3 | \( 1 + (21.8 - 18.3i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (-33.2 - 12.0i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (42.9 - 74.3i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (299. + 519. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-417. - 350. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-252. - 1.43e3i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (3.44e3 - 1.25e3i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-17.7 + 100. i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-440. + 763. i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 8.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-8.16e3 + 6.85e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-4.71e3 - 1.71e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-1.41e3 + 7.99e3i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (2.43e4 - 8.85e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-6.27e3 - 3.56e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (3.69e4 - 1.34e4i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (2.37e3 - 1.34e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-7.02e4 - 2.55e4i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (4.52e4 - 3.79e4i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-4.60e3 + 3.86e3i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (3.65e4 - 6.33e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.17e4 - 9.83e3i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (1.86e4 + 1.05e5i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−16.05647472830621698978720744265, −15.41896408312396122093067988367, −13.92204679374036238051026927137, −12.41142871865722518464186480428, −11.00946646762968178707412264596, −10.06901164994638638453787945418, −8.670546392957449832655225533829, −6.04053258925683381890321476513, −5.85372886937838220383939523656, −3.98122709717672794150443012095,
0.26294002770134863496618480389, 1.91130644582780361871557749589, 4.92413812886504463871848067500, 6.25314395270238521947660400216, 7.66129883674310502719481697168, 9.968847830273283850070351995798, 10.91694778546953193727221195180, 12.26180824624591569054431272149, 12.93919713368812620470458817642, 13.81930969910419765818092088549
