| L(s) = 1 | − 2-s + 2.69·3-s + 4-s + 2.49·5-s − 2.69·6-s − 1.60·7-s − 8-s + 4.24·9-s − 2.49·10-s + 2.04·11-s + 2.69·12-s + 1.60·14-s + 6.71·15-s + 16-s − 4.54·17-s − 4.24·18-s − 4.85·19-s + 2.49·20-s − 4.31·21-s − 2.04·22-s + 2.71·23-s − 2.69·24-s + 1.21·25-s + 3.35·27-s − 1.60·28-s − 9.20·29-s − 6.71·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.55·3-s + 0.5·4-s + 1.11·5-s − 1.09·6-s − 0.606·7-s − 0.353·8-s + 1.41·9-s − 0.788·10-s + 0.617·11-s + 0.777·12-s + 0.428·14-s + 1.73·15-s + 0.250·16-s − 1.10·17-s − 1.00·18-s − 1.11·19-s + 0.557·20-s − 0.942·21-s − 0.436·22-s + 0.565·23-s − 0.549·24-s + 0.243·25-s + 0.646·27-s − 0.303·28-s − 1.70·29-s − 1.22·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742006079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742006079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 - 2.04T + 11T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 + 9.20T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 0.219T + 47T^{2} \) |
| 53 | \( 1 + 2.71T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 1.28T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 5.32T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 4.64T + 97T^{2} \) |
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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
−11.27544072861475689311619612923, −10.21169053678005845976424476082, −9.271315054689697286832554069889, −9.124183720497808064025075784833, −8.032097583245981141471144508013, −6.89531212031146585363038881931, −6.03626399448508789783736714976, −4.12473728616083118300334546827, −2.75664713130300424803391274565, −1.86995629665135213307745984822,
1.86995629665135213307745984822, 2.75664713130300424803391274565, 4.12473728616083118300334546827, 6.03626399448508789783736714976, 6.89531212031146585363038881931, 8.032097583245981141471144508013, 9.124183720497808064025075784833, 9.271315054689697286832554069889, 10.21169053678005845976424476082, 11.27544072861475689311619612923
