| L(s) = 1 | + 0.605·2-s − 1.63·4-s + 3.41·5-s − 3.63·7-s − 2.19·8-s + 2.06·10-s + 2.80·11-s + 5.56·13-s − 2.19·14-s + 1.93·16-s + 5.89·17-s + 4.63·19-s − 5.57·20-s + 1.69·22-s − 6.49·23-s + 6.63·25-s + 3.37·26-s + 5.93·28-s + 3.08·29-s − 1.93·31-s + 5.57·32-s + 3.56·34-s − 12.3·35-s + 6.69·37-s + 2.80·38-s − 7.50·40-s − 11.1·41-s + ⋯ |
| L(s) = 1 | + 0.428·2-s − 0.816·4-s + 1.52·5-s − 1.37·7-s − 0.777·8-s + 0.653·10-s + 0.845·11-s + 1.54·13-s − 0.587·14-s + 0.483·16-s + 1.42·17-s + 1.06·19-s − 1.24·20-s + 0.362·22-s − 1.35·23-s + 1.32·25-s + 0.661·26-s + 1.12·28-s + 0.573·29-s − 0.347·31-s + 0.984·32-s + 0.611·34-s − 2.09·35-s + 1.10·37-s + 0.455·38-s − 1.18·40-s − 1.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.852015569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.852015569\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 - 0.605T + 2T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 2.80T + 11T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 + 1.93T + 31T^{2} \) |
| 37 | \( 1 - 6.69T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 7.20T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 + 6.83T + 61T^{2} \) |
| 71 | \( 1 + 4.01T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19374125063723387700505910765, −9.792270757001724909632995049159, −9.200782219531507168199240098817, −8.247264939769757382841622637416, −6.60462913409141729266584263925, −5.99586261051310211947910322487, −5.40929883810056246711403615520, −3.84009007023817333499633320521, −3.13133607083973442576829667318, −1.27458105779505919429872624615,
1.27458105779505919429872624615, 3.13133607083973442576829667318, 3.84009007023817333499633320521, 5.40929883810056246711403615520, 5.99586261051310211947910322487, 6.60462913409141729266584263925, 8.247264939769757382841622637416, 9.200782219531507168199240098817, 9.792270757001724909632995049159, 10.19374125063723387700505910765
