| L(s) = 1 | − 2-s − 0.920·3-s + 4-s + 2.96·5-s + 0.920·6-s − 7-s − 8-s − 2.15·9-s − 2.96·10-s − 3.73·11-s − 0.920·12-s − 4.18·13-s + 14-s − 2.72·15-s + 16-s + 5.64·17-s + 2.15·18-s + 1.59·19-s + 2.96·20-s + 0.920·21-s + 3.73·22-s + 4.99·23-s + 0.920·24-s + 3.79·25-s + 4.18·26-s + 4.74·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.531·3-s + 0.5·4-s + 1.32·5-s + 0.375·6-s − 0.377·7-s − 0.353·8-s − 0.717·9-s − 0.937·10-s − 1.12·11-s − 0.265·12-s − 1.16·13-s + 0.267·14-s − 0.704·15-s + 0.250·16-s + 1.36·17-s + 0.507·18-s + 0.366·19-s + 0.663·20-s + 0.200·21-s + 0.796·22-s + 1.04·23-s + 0.187·24-s + 0.758·25-s + 0.821·26-s + 0.912·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9976254970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9976254970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 0.920T + 3T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 17 | \( 1 - 5.64T + 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 + 0.281T + 29T^{2} \) |
| 31 | \( 1 + 0.595T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 2.59T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 0.827T + 67T^{2} \) |
| 71 | \( 1 + 8.62T + 71T^{2} \) |
| 73 | \( 1 - 0.950T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−7.916856999538760020684517529748, −7.52772807667328461064140314847, −6.56376642632769960381576103998, −5.96238419418275659075028993329, −5.30765161367469607354868489208, −4.93303901385811151359528880417, −3.15165801830255993486319895610, −2.74184244657271168229778284440, −1.76931969567070417748719544964, −0.58436541222449823467147406233,
0.58436541222449823467147406233, 1.76931969567070417748719544964, 2.74184244657271168229778284440, 3.15165801830255993486319895610, 4.93303901385811151359528880417, 5.30765161367469607354868489208, 5.96238419418275659075028993329, 6.56376642632769960381576103998, 7.52772807667328461064140314847, 7.916856999538760020684517529748
