| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 4.64·11-s − 0.645·13-s + 16-s + 3.29·17-s + 4.29·19-s − 20-s + 4.64·22-s − 3·23-s + 25-s + 0.645·26-s − 2·31-s − 32-s − 3.29·34-s − 5.93·37-s − 4.29·38-s + 40-s + 1.35·41-s + 11.2·43-s − 4.64·44-s + 3·46-s − 9.58·47-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.40·11-s − 0.179·13-s + 0.250·16-s + 0.798·17-s + 0.984·19-s − 0.223·20-s + 0.990·22-s − 0.625·23-s + 0.200·25-s + 0.126·26-s − 0.359·31-s − 0.176·32-s − 0.564·34-s − 0.976·37-s − 0.696·38-s + 0.158·40-s + 0.211·41-s + 1.72·43-s − 0.700·44-s + 0.442·46-s − 1.39·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8886251223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8886251223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 + 0.645T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 9.58T + 47T^{2} \) |
| 53 | \( 1 - 0.291T + 53T^{2} \) |
| 59 | \( 1 + 9.29T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.136849182982014457976106887619, −7.74410788786015766715830926302, −7.26109674022931713265655303244, −6.21018311684457643895745762607, −5.44553828256478317633364518976, −4.75945328574251872505089478810, −3.54974935081577372400791081908, −2.88301552095460879657484639749, −1.85169748182741527303104406663, −0.57567148317144328754404642758,
0.57567148317144328754404642758, 1.85169748182741527303104406663, 2.88301552095460879657484639749, 3.54974935081577372400791081908, 4.75945328574251872505089478810, 5.44553828256478317633364518976, 6.21018311684457643895745762607, 7.26109674022931713265655303244, 7.74410788786015766715830926302, 8.136849182982014457976106887619
