| L(s) = 1 | − 6.40i·3-s + 47.0·5-s − 32.2·7-s + 40.0·9-s + 32.3·11-s − 258. i·13-s − 301. i·15-s + 35.6·17-s + (147. + 329. i)19-s + 206. i·21-s + 5.36·23-s + 1.59e3·25-s − 774. i·27-s + 838. i·29-s − 704. i·31-s + ⋯ |
| L(s) = 1 | − 0.711i·3-s + 1.88·5-s − 0.657·7-s + 0.494·9-s + 0.267·11-s − 1.53i·13-s − 1.33i·15-s + 0.123·17-s + (0.409 + 0.912i)19-s + 0.467i·21-s + 0.0101·23-s + 2.54·25-s − 1.06i·27-s + 0.996i·29-s − 0.733i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.865881034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865881034\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-147. - 329. i)T \) |
| good | 3 | \( 1 + 6.40iT - 81T^{2} \) |
| 5 | \( 1 - 47.0T + 625T^{2} \) |
| 7 | \( 1 + 32.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 32.3T + 1.46e4T^{2} \) |
| 13 | \( 1 + 258. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 35.6T + 8.35e4T^{2} \) |
| 23 | \( 1 - 5.36T + 2.79e5T^{2} \) |
| 29 | \( 1 - 838. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 704. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.26e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.36e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.69e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 4.03e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 2.00e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.43e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 2.95e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 6.54e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.33e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.87e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.80e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 6.93e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.03e4iT - 8.85e7T^{2} \) |
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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
−10.52002819488111234027576452786, −10.03537992650906913555992519887, −9.225555379216639919362340541632, −7.948602097836221787811585604783, −6.79335910212705795654435257918, −6.03661999224741166129305482875, −5.21696590300138674341034471655, −3.26522705558441431517794009719, −2.02083951936309318650136146929, −0.966395800945692440362159699223,
1.42066764849123971167024584508, 2.62984633258144794209783256278, 4.16892113068182824781966262429, 5.24331085837105596692354320223, 6.33222295082609910320698646026, 7.02299646414165081918669532565, 9.038709376448995686956760484095, 9.409165068306034739054374291533, 10.07021806674807368760943904921, 10.96451515638934343890366388333
