L(s) = 1 | − 2.20i·5-s + i·7-s − 0.794·11-s + 5.12·13-s − 0.620i·17-s − 4i·19-s + 0.794·23-s + 0.123·25-s + 3.00i·29-s + 6.24i·31-s + 2.20·35-s + 11.1·37-s − 3.44i·41-s − 4i·43-s − 12.9·47-s + ⋯ |
L(s) = 1 | − 0.987i·5-s + 0.377i·7-s − 0.239·11-s + 1.42·13-s − 0.150i·17-s − 0.917i·19-s + 0.165·23-s + 0.0246·25-s + 0.557i·29-s + 1.12i·31-s + 0.373·35-s + 1.82·37-s − 0.538i·41-s − 0.609i·43-s − 1.88·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.984789884\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.984789884\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 5 | \( 1 + 2.20iT - 5T^{2} \) |
| 11 | \( 1 + 0.794T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 0.620iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 0.794T + 23T^{2} \) |
| 29 | \( 1 - 3.00iT - 29T^{2} \) |
| 31 | \( 1 - 6.24iT - 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 3.00iT - 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 5.12iT - 67T^{2} \) |
| 71 | \( 1 - 8.03T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 11.3iT - 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 2.20iT - 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−8.457217522292583676345440006018, −7.78466624466667835662985901786, −6.75188849087867554635704721062, −6.12633153401097284299040270990, −5.18481135765589686726674849957, −4.75525235631373661430450016691, −3.71837068897237738702017402258, −2.85332183506924646546788812807, −1.65055234328606156615737896406, −0.70914409020238024244004452469,
1.00897253779843828827818059456, 2.19712465555702103517502153411, 3.19881341621709906031824415065, 3.82028946789270103474143499287, 4.68558312673307324548018693218, 5.91467307406879652276158370697, 6.25889658774309227486509427068, 7.02878775825088321031729758044, 8.004012667076370785654502301942, 8.214646458123272358517523407195