| L(s) = 1 | + (−1.68 − 1.08i)2-s + (−0.719 − 5.00i)3-s + (1.66 + 3.63i)4-s + (−4.79 − 1.40i)5-s + (−4.20 + 9.19i)6-s + (9.79 − 11.3i)7-s + (1.13 − 7.91i)8-s + (1.38 − 0.406i)9-s + (6.54 + 7.55i)10-s + (33.3 − 21.4i)11-s + (17.0 − 10.9i)12-s + (−17.7 − 20.4i)13-s + (−28.7 + 8.43i)14-s + (−3.59 + 25.0i)15-s + (−10.4 + 12.0i)16-s + (1.56 − 3.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.138 − 0.962i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (−0.285 + 0.625i)6-s + (0.529 − 0.610i)7-s + (0.0503 − 0.349i)8-s + (0.0513 − 0.0150i)9-s + (0.207 + 0.238i)10-s + (0.914 − 0.588i)11-s + (0.409 − 0.262i)12-s + (−0.377 − 0.436i)13-s + (−0.548 + 0.160i)14-s + (−0.0619 + 0.430i)15-s + (−0.163 + 0.188i)16-s + (0.0222 − 0.0487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0256280 - 1.00138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0256280 - 1.00138i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.68 + 1.08i)T \) |
| 5 | \( 1 + (4.79 + 1.40i)T \) |
| 23 | \( 1 + (-71.3 + 84.1i)T \) |
| good | 3 | \( 1 + (0.719 + 5.00i)T + (-25.9 + 7.60i)T^{2} \) |
| 7 | \( 1 + (-9.79 + 11.3i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-33.3 + 21.4i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (17.7 + 20.4i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 3.41i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (38.8 + 85.0i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (122. - 268. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (34.3 - 238. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-153. + 45.1i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (473. + 139. i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (40.6 + 282. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (341. - 393. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (188. + 217. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-114. + 793. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-752. - 483. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-120. - 77.2i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (431. + 945. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (113. + 130. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-158. + 46.4i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-76.4 - 531. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-1.47e3 - 433. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−11.23335367571967163542587378354, −10.58639643633985330368941617501, −9.147729833968580031450436943526, −8.312681550883887427045882152208, −7.22749378652856588783807222984, −6.67553266195173843491534268042, −4.83577832363008532818909749743, −3.38422380377534434604592190944, −1.61506386694699440445691211993, −0.53362441492426335649476685416,
1.83148269133468222316485018485, 3.89072697953292076015339999603, 4.86638436999505821337306474622, 6.13358300273697235863147474334, 7.38811063064717891663487320738, 8.357801662863595784384918646814, 9.547357062091408526381828312239, 9.927637022017107821317377784800, 11.34302390839050287307775757949, 11.71975692330704154387790307020
