| L(s) = 1 | − i·2-s − 4-s + 3.34i·5-s + (2.62 + 0.292i)7-s + i·8-s + 3.34·10-s + 5.17·11-s + 4.32·13-s + (0.292 − 2.62i)14-s + 16-s − 0.584i·17-s + (1.47 + 4.10i)19-s − 3.34i·20-s − 5.17i·22-s + 3.25·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.49i·5-s + (0.993 + 0.110i)7-s + 0.353i·8-s + 1.05·10-s + 1.56·11-s + 1.20·13-s + (0.0781 − 0.702i)14-s + 0.250·16-s − 0.141i·17-s + (0.337 + 0.941i)19-s − 0.747i·20-s − 1.10i·22-s + 0.679·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.300853730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300853730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 - 0.292i)T \) |
| 19 | \( 1 + (-1.47 - 4.10i)T \) |
| good | 5 | \( 1 - 3.34iT - 5T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 + 0.584iT - 17T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 + 9.37iT - 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 - 4.23iT - 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 - 5.75iT - 47T^{2} \) |
| 53 | \( 1 - 4.08iT - 53T^{2} \) |
| 59 | \( 1 - 2.75T + 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 13.6iT - 67T^{2} \) |
| 71 | \( 1 + 11.6iT - 71T^{2} \) |
| 73 | \( 1 + 3.07iT - 73T^{2} \) |
| 79 | \( 1 - 10.2iT - 79T^{2} \) |
| 83 | \( 1 - 2.68iT - 83T^{2} \) |
| 89 | \( 1 - 7.61T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−9.133663930036364839603626282713, −8.242984326338959590793756286125, −7.57070479416604000320506329157, −6.51025229571446777317416047866, −6.08254332993493615713703350003, −4.86233764275004197519964941038, −3.78426410482727795154072701492, −3.38195054067768257591911192287, −2.12202787722154549770945294503, −1.28075015038763151820501366069,
0.999311627033494224895790897953, 1.56656875875285288556257802293, 3.58606559918017879205687683846, 4.29639914893274130787503505878, 5.09519872348304027185900840996, 5.61217661580822883484341472326, 6.73678330753116337522276008820, 7.32658944405987511725831114919, 8.557812436839412187521090355360, 8.728809883504560541463304014129
