L(s) = 1 | − 1.45·2-s + 0.119·4-s − 2.38·5-s − 7-s + 2.73·8-s + 3.47·10-s − 0.911·13-s + 1.45·14-s − 4.22·16-s + 2.18·17-s + 0.711·19-s − 0.285·20-s + 7.80·23-s + 0.698·25-s + 1.32·26-s − 0.119·28-s − 8.86·29-s − 6.62·31-s + 0.675·32-s − 3.18·34-s + 2.38·35-s − 4.20·37-s − 1.03·38-s − 6.53·40-s + 11.9·41-s + 7.51·43-s − 11.3·46-s + ⋯ |
L(s) = 1 | − 1.02·2-s + 0.0598·4-s − 1.06·5-s − 0.377·7-s + 0.967·8-s + 1.09·10-s − 0.252·13-s + 0.389·14-s − 1.05·16-s + 0.530·17-s + 0.163·19-s − 0.0638·20-s + 1.62·23-s + 0.139·25-s + 0.260·26-s − 0.0226·28-s − 1.64·29-s − 1.19·31-s + 0.119·32-s − 0.545·34-s + 0.403·35-s − 0.691·37-s − 0.168·38-s − 1.03·40-s + 1.86·41-s + 1.14·43-s − 1.67·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5011162347\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5011162347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 13 | \( 1 + 0.911T + 13T^{2} \) |
| 17 | \( 1 - 2.18T + 17T^{2} \) |
| 19 | \( 1 - 0.711T + 19T^{2} \) |
| 23 | \( 1 - 7.80T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 7.51T + 43T^{2} \) |
| 47 | \( 1 + 6.27T + 47T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 - 9.62T + 59T^{2} \) |
| 61 | \( 1 + 0.0627T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 + 4.38T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 0.261T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−7.73095395624374122080173828001, −7.45324710477697369747599340173, −6.95215029912810885454184068770, −5.74631074076542939348774610369, −5.09039809734285007109952613905, −4.12693570707946889029657033695, −3.64554719061296354498366202123, −2.62226773967796975621425380241, −1.42677117157122029804410031917, −0.44435449608172592586083596473,
0.44435449608172592586083596473, 1.42677117157122029804410031917, 2.62226773967796975621425380241, 3.64554719061296354498366202123, 4.12693570707946889029657033695, 5.09039809734285007109952613905, 5.74631074076542939348774610369, 6.95215029912810885454184068770, 7.45324710477697369747599340173, 7.73095395624374122080173828001