| L(s) = 1 | + 2.82i·2-s + (1.26 + 8.91i)3-s − 8.00·4-s + 33.9i·5-s + (−25.2 + 3.58i)6-s − 68.2·7-s − 22.6i·8-s + (−77.7 + 22.5i)9-s − 96.1·10-s − 130. i·11-s + (−10.1 − 71.2i)12-s + 135.·13-s − 193. i·14-s + (−302. + 43.0i)15-s + 64.0·16-s + 82.7i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.140 + 0.990i)3-s − 0.500·4-s + 1.35i·5-s + (−0.700 + 0.0995i)6-s − 1.39·7-s − 0.353i·8-s + (−0.960 + 0.278i)9-s − 0.961·10-s − 1.07i·11-s + (−0.0703 − 0.495i)12-s + 0.803·13-s − 0.984i·14-s + (−1.34 + 0.191i)15-s + 0.250·16-s + 0.286i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5335523031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5335523031\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + (-1.26 - 8.91i)T \) |
| 29 | \( 1 + 156. iT \) |
| good | 5 | \( 1 - 33.9iT - 625T^{2} \) |
| 7 | \( 1 + 68.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 130. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 135.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 82.7iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 162.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 973. iT - 2.79e5T^{2} \) |
| 31 | \( 1 + 840.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 827.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.58e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.60e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.18e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.71e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.58e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 964.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.73e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.85e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.86e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 9.42e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.19e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.68e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.20e3T + 8.85e7T^{2} \) |
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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
−13.24042838343757133077540086545, −11.48233246213672293778712884530, −10.66266865349990605636832991258, −9.745948605130950232762180042582, −8.915009025203280813771567351042, −7.55418067046201319561797821562, −6.32744016278274563738479719826, −5.67567452066750373439145635304, −3.63144021144816653364209485330, −3.19805751870641316890939175938,
0.20183522399009206186194368842, 1.41335832398750991660979232924, 2.91608916625861045905129341951, 4.45132533267305993361501644919, 5.88176989730106945190707286161, 7.06740243260789632619534985818, 8.466083962521044349485416328684, 9.164825609687866772557404926150, 10.15189572138121254626884623917, 11.67053917528102595453632097127
