| L(s) = 1 | + 0.232·2-s − 3·3-s − 7.94·4-s − 7.75·5-s − 0.698·6-s − 3.71·8-s + 9·9-s − 1.80·10-s + 11·11-s + 23.8·12-s − 59.4·13-s + 23.2·15-s + 62.7·16-s + 45.4·17-s + 2.09·18-s − 111.·19-s + 61.6·20-s + 2.56·22-s + 105.·23-s + 11.1·24-s − 64.8·25-s − 13.8·26-s − 27·27-s + 10.0·29-s + 5.42·30-s − 315.·31-s + 44.3·32-s + ⋯ |
| L(s) = 1 | + 0.0823·2-s − 0.577·3-s − 0.993·4-s − 0.693·5-s − 0.0475·6-s − 0.164·8-s + 0.333·9-s − 0.0571·10-s + 0.301·11-s + 0.573·12-s − 1.26·13-s + 0.400·15-s + 0.979·16-s + 0.648·17-s + 0.0274·18-s − 1.34·19-s + 0.689·20-s + 0.0248·22-s + 0.954·23-s + 0.0947·24-s − 0.518·25-s − 0.104·26-s − 0.192·27-s + 0.0641·29-s + 0.0329·30-s − 1.82·31-s + 0.244·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2725971284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2725971284\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 0.232T + 8T^{2} \) |
| 5 | \( 1 + 7.75T + 125T^{2} \) |
| 13 | \( 1 + 59.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 10.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 315.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 487.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 358.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 891.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 654.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 346.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 774.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 502.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 939.T + 9.12e5T^{2} \) |
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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
−8.980471106325959807797347152400, −8.309759834553664971151658601789, −7.40483218261997931770810714277, −6.70323195826589254754403568191, −5.48269127112848088470597370924, −4.94761121596129519877875184473, −4.08029439986823752945027354355, −3.30529450414883782860866138537, −1.71852631106716434987633588371, −0.25550758689102198199491607301,
0.25550758689102198199491607301, 1.71852631106716434987633588371, 3.30529450414883782860866138537, 4.08029439986823752945027354355, 4.94761121596129519877875184473, 5.48269127112848088470597370924, 6.70323195826589254754403568191, 7.40483218261997931770810714277, 8.309759834553664971151658601789, 8.980471106325959807797347152400
