| L(s) = 1 | − 2.23i·5-s + 3.87i·7-s + 1.73·11-s − 2·13-s − 4.47i·17-s + 6.92·23-s + 4.47i·29-s − 3.87i·31-s + 8.66·35-s + 4·37-s + 8.94i·41-s + 7.74i·43-s + 3.46·47-s − 8.00·49-s − 2.23i·53-s + ⋯ |
| L(s) = 1 | − 0.999i·5-s + 1.46i·7-s + 0.522·11-s − 0.554·13-s − 1.08i·17-s + 1.44·23-s + 0.830i·29-s − 0.695i·31-s + 1.46·35-s + 0.657·37-s + 1.39i·41-s + 1.18i·43-s + 0.505·47-s − 1.14·49-s − 0.307i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.757916897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757916897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.23iT - 5T^{2} \) |
| 7 | \( 1 - 3.87iT - 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 3.87iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 8.94iT - 41T^{2} \) |
| 43 | \( 1 - 7.74iT - 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.23iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 7.74iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 7.74iT - 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 4.47iT - 89T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
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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
−9.295907128106874251754033068989, −8.715245056520014583456357927119, −7.901930991420573274841940002160, −6.89300381961172343876538474741, −5.99545367709765374992708686670, −5.03706830281016540122094677830, −4.74006368677779757328994762920, −3.20812438538059804743906551918, −2.30953153934431804660223957894, −0.972589239802239879566466175296,
0.912784669434568233243060684461, 2.34910766146292448620771132608, 3.52744450935604959873829794120, 4.09474941180624161284134508973, 5.20441920637936385286345674552, 6.38919823661287754775416674235, 7.00220041392202997163361613404, 7.46954256416573118190085433158, 8.485911910800638601791973140238, 9.432189167915631793471249139414
