L(s) = 1 | − 5-s − 1.56·7-s − 2·11-s + 0.561·13-s + 1.56·17-s + 6·19-s − 23-s + 25-s + 2.12·29-s − 9.24·31-s + 1.56·35-s − 0.438·37-s + 4.12·41-s + 7.68·47-s − 4.56·49-s + 0.438·53-s + 2·55-s − 8.68·59-s + 1.12·61-s − 0.561·65-s − 4.43·67-s − 1.87·71-s − 8.56·73-s + 3.12·77-s + 13.1·79-s − 14.9·83-s − 1.56·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.590·7-s − 0.603·11-s + 0.155·13-s + 0.378·17-s + 1.37·19-s − 0.208·23-s + 0.200·25-s + 0.394·29-s − 1.66·31-s + 0.263·35-s − 0.0720·37-s + 0.643·41-s + 1.12·47-s − 0.651·49-s + 0.0602·53-s + 0.269·55-s − 1.13·59-s + 0.143·61-s − 0.0696·65-s − 0.542·67-s − 0.222·71-s − 1.00·73-s + 0.355·77-s + 1.47·79-s − 1.63·83-s − 0.169·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 + 0.438T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 - 0.438T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 + 4.43T + 67T^{2} \) |
| 71 | \( 1 + 1.87T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 2.24T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−7.82142074295995730408799812098, −7.50417492171943738086114574144, −6.65115116690813656959056279803, −5.73940719674188407132664032847, −5.18359000576561728478540914690, −4.13923595809106487854865692892, −3.36480036572835287791980102643, −2.65496218881759201842139255484, −1.31189355048478474088904992157, 0,
1.31189355048478474088904992157, 2.65496218881759201842139255484, 3.36480036572835287791980102643, 4.13923595809106487854865692892, 5.18359000576561728478540914690, 5.73940719674188407132664032847, 6.65115116690813656959056279803, 7.50417492171943738086114574144, 7.82142074295995730408799812098