L(s) = 1 | + 2-s + 1.23·3-s + 4-s + 0.483·5-s + 1.23·6-s + 7-s + 8-s − 1.47·9-s + 0.483·10-s − 0.207·11-s + 1.23·12-s + 4.81·13-s + 14-s + 0.597·15-s + 16-s + 6.97·17-s − 1.47·18-s − 3.74·19-s + 0.483·20-s + 1.23·21-s − 0.207·22-s − 2.03·23-s + 1.23·24-s − 4.76·25-s + 4.81·26-s − 5.52·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.713·3-s + 0.5·4-s + 0.216·5-s + 0.504·6-s + 0.377·7-s + 0.353·8-s − 0.491·9-s + 0.152·10-s − 0.0625·11-s + 0.356·12-s + 1.33·13-s + 0.267·14-s + 0.154·15-s + 0.250·16-s + 1.69·17-s − 0.347·18-s − 0.859·19-s + 0.108·20-s + 0.269·21-s − 0.0442·22-s − 0.425·23-s + 0.252·24-s − 0.953·25-s + 0.943·26-s − 1.06·27-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
\(L(1)\) |
\(\approx\) |
\(4.844708826\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.844708826\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 - 0.483T + 5T^{2} \) |
| 11 | \( 1 + 0.207T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 1.08T + 29T^{2} \) |
| 31 | \( 1 - 7.69T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 + 2.15T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 + 1.64T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 + 7.86T + 79T^{2} \) |
| 83 | \( 1 - 4.48T + 83T^{2} \) |
| 89 | \( 1 + 5.10T + 89T^{2} \) |
| 97 | \( 1 - 9.35T + 97T^{2} \) |
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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
−8.031797759485074756201990189859, −7.57451719380198446746520859349, −6.39832531473143324830490016122, −5.89438005941209945095084835176, −5.33791924861222469729561314223, −4.19063398320255715165170532600, −3.70576042925809841533237168172, −2.86050409789472359050594530468, −2.09979804589005090759587802197, −1.06034089898346341506337680806,
1.06034089898346341506337680806, 2.09979804589005090759587802197, 2.86050409789472359050594530468, 3.70576042925809841533237168172, 4.19063398320255715165170532600, 5.33791924861222469729561314223, 5.89438005941209945095084835176, 6.39832531473143324830490016122, 7.57451719380198446746520859349, 8.031797759485074756201990189859