| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.343759500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343759500\) |
| \(L(1)\) |
|
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 | \( 1 - T^{2} \) |
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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
−8.801145493178081158006207374066, −8.261909918631340658512611404298, −7.19775576067578459217102556709, −6.26143453957551690132687220860, −5.42404504590758897658248208246, −5.04665444579986386678841395121, −4.88907013925501183638957443111, −3.01914574155887625394628549148, −2.39177293971338216196000008508, −1.05778027627910879612288996359,
1.05778027627910879612288996359, 2.39177293971338216196000008508, 3.01914574155887625394628549148, 4.88907013925501183638957443111, 5.04665444579986386678841395121, 5.42404504590758897658248208246, 6.26143453957551690132687220860, 7.19775576067578459217102556709, 8.261909918631340658512611404298, 8.801145493178081158006207374066
