| L(s) = 1 | + 1.77·2-s + 1.82·3-s + 1.15·4-s + 1.34·5-s + 3.23·6-s − 1.23·7-s − 1.50·8-s + 0.320·9-s + 2.39·10-s − 11-s + 2.10·12-s + 6.16·13-s − 2.19·14-s + 2.45·15-s − 4.97·16-s − 5.62·17-s + 0.569·18-s + 4.19·19-s + 1.55·20-s − 2.25·21-s − 1.77·22-s − 2.75·23-s − 2.74·24-s − 3.18·25-s + 10.9·26-s − 4.88·27-s − 1.42·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 1.05·3-s + 0.576·4-s + 0.603·5-s + 1.32·6-s − 0.467·7-s − 0.531·8-s + 0.106·9-s + 0.757·10-s − 0.301·11-s + 0.606·12-s + 1.70·13-s − 0.586·14-s + 0.634·15-s − 1.24·16-s − 1.36·17-s + 0.134·18-s + 0.961·19-s + 0.347·20-s − 0.491·21-s − 0.378·22-s − 0.574·23-s − 0.559·24-s − 0.636·25-s + 2.14·26-s − 0.939·27-s − 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.116407664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.116407664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 - 6.16T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 + 2.75T + 23T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 37 | \( 1 - 0.867T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 - 7.77T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 + 1.75T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2.40T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 - 8.77T + 97T^{2} \) |
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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
−11.71046800112327240747309012240, −10.79109875562377258916012217721, −9.342460416112551951367118356406, −8.960088533316565601567549470498, −7.73284636317464451416549847622, −6.26693184902434469025114789419, −5.71302076514270468171449904651, −4.19976776244906528898070159301, −3.34798479625447114870128448526, −2.27471665333969987914413801905,
2.27471665333969987914413801905, 3.34798479625447114870128448526, 4.19976776244906528898070159301, 5.71302076514270468171449904651, 6.26693184902434469025114789419, 7.73284636317464451416549847622, 8.960088533316565601567549470498, 9.342460416112551951367118356406, 10.79109875562377258916012217721, 11.71046800112327240747309012240
