| L(s) = 1 | + 13.8i·3-s + 16.6·5-s + 34.1·7-s − 111.·9-s + 198.·11-s − 164. i·13-s + 230. i·15-s + 463.·17-s + (176. − 314. i)19-s + 472. i·21-s + 807.·23-s − 348.·25-s − 417. i·27-s − 424. i·29-s − 1.28e3i·31-s + ⋯ |
| L(s) = 1 | + 1.54i·3-s + 0.665·5-s + 0.695·7-s − 1.37·9-s + 1.64·11-s − 0.974i·13-s + 1.02i·15-s + 1.60·17-s + (0.489 − 0.872i)19-s + 1.07i·21-s + 1.52·23-s − 0.557·25-s − 0.573i·27-s − 0.504i·29-s − 1.33i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.880117412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.880117412\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-176. + 314. i)T \) |
| good | 3 | \( 1 - 13.8iT - 81T^{2} \) |
| 5 | \( 1 - 16.6T + 625T^{2} \) |
| 7 | \( 1 - 34.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 198.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 164. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 463.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 807.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 424. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.28e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.12e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 408. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.54e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 198.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.51e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.47e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.11e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.50e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 4.05e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 7.79T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.34e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.74e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 8.13e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 70.4iT - 8.85e7T^{2} \) |
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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
−11.16161971022078541096620035938, −10.09704822275170680594859074435, −9.536128184995854050726730601467, −8.762527227129314419816616195206, −7.47047250390070002060438926075, −5.92373826815601201457389935854, −5.15273203125900794473861891210, −4.09641082864021276991647797029, −3.02838444563472862756916934770, −1.15011979838501128754420549949,
1.42327243812986356473225739464, 1.50048287938005592241260992336, 3.36225845890272235848080774957, 5.09547999496034848584774704812, 6.27127887400137080209261892723, 6.93166449423092599570367373792, 7.899783137228306784819626145950, 8.903049663250940146699281439490, 9.843551609132327458042037619611, 11.29412364601390673161983980446
