| L(s) = 1 | + 0.922i·5-s + (2.06 − 1.65i)7-s − 1.61i·11-s − 2.82i·13-s − 0.922i·17-s − 0.540·19-s + 7.05i·23-s + 4.14·25-s + 1.01·29-s + 8.67·31-s + (1.52 + 1.90i)35-s − 8.19·37-s − 6.57i·41-s − 4.96i·43-s − 2.60·47-s + ⋯ |
| L(s) = 1 | + 0.412i·5-s + (0.781 − 0.624i)7-s − 0.487i·11-s − 0.784i·13-s − 0.223i·17-s − 0.123·19-s + 1.47i·23-s + 0.829·25-s + 0.188·29-s + 1.55·31-s + (0.257 + 0.322i)35-s − 1.34·37-s − 1.02i·41-s − 0.756i·43-s − 0.380·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.020891751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020891751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.06 + 1.65i)T \) |
| good | 5 | \( 1 - 0.922iT - 5T^{2} \) |
| 11 | \( 1 + 1.61iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 0.922iT - 17T^{2} \) |
| 19 | \( 1 + 0.540T + 19T^{2} \) |
| 23 | \( 1 - 7.05iT - 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 - 8.67T + 31T^{2} \) |
| 37 | \( 1 + 8.19T + 37T^{2} \) |
| 41 | \( 1 + 6.57iT - 41T^{2} \) |
| 43 | \( 1 + 4.96iT - 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 7.01iT - 61T^{2} \) |
| 67 | \( 1 + 6.80iT - 67T^{2} \) |
| 71 | \( 1 - 5.21iT - 71T^{2} \) |
| 73 | \( 1 + 7.50iT - 73T^{2} \) |
| 79 | \( 1 + 7.11iT - 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 + 17.9iT - 89T^{2} \) |
| 97 | \( 1 + 4.18iT - 97T^{2} \) |
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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
−8.434406602536183998658316322174, −7.42135637605728059076122258216, −7.14440957916450134377061554468, −6.07364452014589615686160044276, −5.34259703006825769199224187368, −4.61354044577478383255264630269, −3.61675376266883623071357125083, −2.95000341789171110598975913949, −1.75095430442060618888204368688, −0.66004958200708868263862903961,
1.10862979939098980544983564277, 2.09166756273381128844239087938, 2.91443496776206031489916915647, 4.30186015113898303727812479420, 4.67490342977742442390766276789, 5.45309963589142578904496590797, 6.47300405397534218696303291186, 6.92898459412559792070618652562, 8.140659451945772696749834118329, 8.416253998243396963585711297210
