| L(s) = 1 | + 10.5i·3-s − 16.8·5-s + 38.6·7-s − 31.1·9-s + 70.8·11-s + 231. i·13-s − 178. i·15-s − 90.4·17-s + (319. + 167. i)19-s + 409. i·21-s + 285.·23-s − 342.·25-s + 527. i·27-s − 285. i·29-s + 852. i·31-s + ⋯ |
| L(s) = 1 | + 1.17i·3-s − 0.672·5-s + 0.788·7-s − 0.384·9-s + 0.585·11-s + 1.36i·13-s − 0.791i·15-s − 0.312·17-s + (0.885 + 0.464i)19-s + 0.928i·21-s + 0.539·23-s − 0.547·25-s + 0.723i·27-s − 0.340i·29-s + 0.886i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.585054445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.585054445\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-319. - 167. i)T \) |
| good | 3 | \( 1 - 10.5iT - 81T^{2} \) |
| 5 | \( 1 + 16.8T + 625T^{2} \) |
| 7 | \( 1 - 38.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 70.8T + 1.46e4T^{2} \) |
| 13 | \( 1 - 231. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 90.4T + 8.35e4T^{2} \) |
| 23 | \( 1 - 285.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 285. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 852. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 54.8iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 37.4iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 691.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.63e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.83e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.33e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.87e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.05e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 3.31e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.54e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.03e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 3.54e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.32e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 4.28e3iT - 8.85e7T^{2} \) |
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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
−11.56296818077690454176183295028, −10.56332637085373613254488771099, −9.555007067191116101331561437931, −8.843058616757090716992253656625, −7.74081402269929674819662238214, −6.63561959456282148604548484281, −5.09382397700190065327372428318, −4.33421363621243373435512734249, −3.48651568414419891908533799216, −1.57273671101456452442317579990,
0.51375288684573235992880054090, 1.61437807415728380898266585573, 3.11579598161688225738112461760, 4.56786324421376733094603909948, 5.81129952625154433448013053949, 7.02583772123600969436860681990, 7.74289353417321726072686916966, 8.387606104427720530086205012131, 9.701901383473917290314894977895, 11.03691686036225385218180594202
