| L(s) = 1 | + (1.32 − 1.11i)3-s − 5-s + 1.85i·7-s + (0.492 − 2.95i)9-s − 5.93·11-s − 1.57·13-s + (−1.32 + 1.11i)15-s − 4.36·17-s + 0.298i·19-s + (2.07 + 2.44i)21-s + (1.25 + 4.62i)23-s + 25-s + (−2.66 − 4.46i)27-s + 6.89i·29-s − 9.74·31-s + ⋯ |
| L(s) = 1 | + (0.762 − 0.646i)3-s − 0.447·5-s + 0.699i·7-s + (0.164 − 0.986i)9-s − 1.78·11-s − 0.436·13-s + (−0.341 + 0.289i)15-s − 1.05·17-s + 0.0684i·19-s + (0.452 + 0.533i)21-s + (0.262 + 0.964i)23-s + 0.200·25-s + (−0.512 − 0.858i)27-s + 1.27i·29-s − 1.75·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1610211724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1610211724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.32 + 1.11i)T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-1.25 - 4.62i)T \) |
| good | 7 | \( 1 - 1.85iT - 7T^{2} \) |
| 11 | \( 1 + 5.93T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 - 0.298iT - 19T^{2} \) |
| 29 | \( 1 - 6.89iT - 29T^{2} \) |
| 31 | \( 1 + 9.74T + 31T^{2} \) |
| 37 | \( 1 - 7.74iT - 37T^{2} \) |
| 41 | \( 1 + 7.49iT - 41T^{2} \) |
| 43 | \( 1 + 4.96iT - 43T^{2} \) |
| 47 | \( 1 + 4.73iT - 47T^{2} \) |
| 53 | \( 1 + 2.15T + 53T^{2} \) |
| 59 | \( 1 + 3.27iT - 59T^{2} \) |
| 61 | \( 1 + 12.6iT - 61T^{2} \) |
| 67 | \( 1 - 13.5iT - 67T^{2} \) |
| 71 | \( 1 - 1.09iT - 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 - 5.00iT - 79T^{2} \) |
| 83 | \( 1 + 5.01T + 83T^{2} \) |
| 89 | \( 1 + 8.95T + 89T^{2} \) |
| 97 | \( 1 + 11.2iT - 97T^{2} \) |
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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
−9.757306161689468204737228803345, −8.822498239989641430175574803851, −8.400982223776981779184910847788, −7.39667756629402658149861734145, −7.03935112695057474286703912909, −5.68951238254917899711576012740, −4.98754953697123120652377481281, −3.63088901877099650118074785970, −2.73972527511015475045525858479, −1.89669963029398987142171232398,
0.05319769730269943486905006307, 2.24078340470908168008969651622, 3.01728291096307961588845491210, 4.22401053644980651620782284060, 4.70360144288499491842952839710, 5.78171862082820361015369213821, 7.19781871305640238416888343078, 7.67288495672102461603790782853, 8.410281401271058748295424719309, 9.255370561721256921145123416681
