| L(s) = 1 | + 2.51·2-s + 0.988·3-s + 4.30·4-s + 0.809·5-s + 2.48·6-s + 2.08·7-s + 5.78·8-s − 2.02·9-s + 2.03·10-s + 0.174·11-s + 4.25·12-s + 1.58·13-s + 5.23·14-s + 0.799·15-s + 5.91·16-s + 2.42·17-s − 5.08·18-s − 0.892·19-s + 3.48·20-s + 2.06·21-s + 0.437·22-s − 4.85·23-s + 5.71·24-s − 4.34·25-s + 3.98·26-s − 4.96·27-s + 8.97·28-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.570·3-s + 2.15·4-s + 0.361·5-s + 1.01·6-s + 0.788·7-s + 2.04·8-s − 0.674·9-s + 0.642·10-s + 0.0525·11-s + 1.22·12-s + 0.440·13-s + 1.40·14-s + 0.206·15-s + 1.47·16-s + 0.587·17-s − 1.19·18-s − 0.204·19-s + 0.778·20-s + 0.449·21-s + 0.0932·22-s − 1.01·23-s + 1.16·24-s − 0.868·25-s + 0.781·26-s − 0.955·27-s + 1.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.636105269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.636105269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1759 | \( 1 - T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 - 0.988T + 3T^{2} \) |
| 5 | \( 1 - 0.809T + 5T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 - 0.174T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 2.42T + 17T^{2} \) |
| 19 | \( 1 + 0.892T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 0.384T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 5.84T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 + 9.61T + 67T^{2} \) |
| 71 | \( 1 - 3.23T + 71T^{2} \) |
| 73 | \( 1 + 3.87T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 1.99T + 97T^{2} \) |
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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
−9.293648831847178721313222694898, −8.196846225010305128140450673162, −7.73726585307130418169690488358, −6.51368636074897312173208421120, −5.87201125111447772508716987218, −5.21094421176340378482966988309, −4.26181097848993732451652865454, −3.50845740879403620623355165554, −2.57198086039079186503557863220, −1.73408558671891566192624580232,
1.73408558671891566192624580232, 2.57198086039079186503557863220, 3.50845740879403620623355165554, 4.26181097848993732451652865454, 5.21094421176340378482966988309, 5.87201125111447772508716987218, 6.51368636074897312173208421120, 7.73726585307130418169690488358, 8.196846225010305128140450673162, 9.293648831847178721313222694898
