| L(s) = 1 | + 1.41·2-s + 1.00·4-s − 0.765·5-s + 9-s − 1.08·10-s + 1.84·11-s − 0.999·16-s − 17-s + 1.41·18-s − 0.765·20-s + 2.61·22-s + 0.765·23-s − 0.414·25-s − 1.84·29-s − 1.84·31-s − 1.41·32-s − 1.41·34-s + 1.00·36-s + 0.765·41-s + 1.84·44-s − 0.765·45-s + 1.08·46-s − 47-s + 49-s − 0.585·50-s − 1.41·55-s − 2.61·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.00·4-s − 0.765·5-s + 9-s − 1.08·10-s + 1.84·11-s − 0.999·16-s − 17-s + 1.41·18-s − 0.765·20-s + 2.61·22-s + 0.765·23-s − 0.414·25-s − 1.84·29-s − 1.84·31-s − 1.41·32-s − 1.41·34-s + 1.00·36-s + 0.765·41-s + 1.84·44-s − 0.765·45-s + 1.08·46-s − 47-s + 49-s − 0.585·50-s − 1.41·55-s − 2.61·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.824218453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824218453\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 0.765T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.84T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.765T + T^{2} \) |
| 29 | \( 1 + 1.84T + T^{2} \) |
| 31 | \( 1 + 1.84T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.765T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.84T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−11.01861918418803294888987320457, −9.406117835751243348406482781202, −9.014245185247014718668730026848, −7.49056534367430194780469798964, −6.86812145389592239968068311951, −6.01721691391608768666327825089, −4.84182717733109462279146146831, −3.95050209520928067550872210290, −3.64542438790764153491915610683, −1.87223478768312454428650563243,
1.87223478768312454428650563243, 3.64542438790764153491915610683, 3.95050209520928067550872210290, 4.84182717733109462279146146831, 6.01721691391608768666327825089, 6.86812145389592239968068311951, 7.49056534367430194780469798964, 9.014245185247014718668730026848, 9.406117835751243348406482781202, 11.01861918418803294888987320457
