| L(s) = 1 | − 0.347·2-s − 0.684·3-s − 0.879·4-s + 1.96·5-s + 0.237·6-s − 7-s + 0.652·8-s − 0.532·9-s − 0.684·10-s − 11-s + 0.601·12-s + 1.73·13-s + 0.347·14-s − 1.34·15-s + 0.652·16-s − 1.28·17-s + 0.184·18-s − 1.73·20-s + 0.684·21-s + 0.347·22-s + 23-s − 0.446·24-s + 2.87·25-s − 0.601·26-s + 1.04·27-s + 0.879·28-s − 1.87·29-s + ⋯ |
| L(s) = 1 | − 0.347·2-s − 0.684·3-s − 0.879·4-s + 1.96·5-s + 0.237·6-s − 7-s + 0.652·8-s − 0.532·9-s − 0.684·10-s − 11-s + 0.601·12-s + 1.73·13-s + 0.347·14-s − 1.34·15-s + 0.652·16-s − 1.28·17-s + 0.184·18-s − 1.73·20-s + 0.684·21-s + 0.347·22-s + 23-s − 0.446·24-s + 2.87·25-s − 0.601·26-s + 1.04·27-s + 0.879·28-s − 1.87·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7893488418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7893488418\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.347T + T^{2} \) |
| 3 | \( 1 + 0.684T + T^{2} \) |
| 5 | \( 1 - 1.96T + T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 + 1.28T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.28T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.684T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| 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
−9.114437117079700269265617770829, −8.440380645133613988518787430808, −7.13069396333858338299453190414, −6.32718592716134258673214753593, −5.62474712373321787914037586107, −5.49650302070396175891813836480, −4.30592251167204352831845041925, −3.12834159359762304708597431312, −2.16954680516121578815263408390, −0.864784803723543020446377103744,
0.864784803723543020446377103744, 2.16954680516121578815263408390, 3.12834159359762304708597431312, 4.30592251167204352831845041925, 5.49650302070396175891813836480, 5.62474712373321787914037586107, 6.32718592716134258673214753593, 7.13069396333858338299453190414, 8.440380645133613988518787430808, 9.114437117079700269265617770829
