| L(s) = 1 | − 2-s + 1.83·3-s + 4-s + 5-s − 1.83·6-s + 2.63·7-s − 8-s + 0.378·9-s − 10-s + 4.43·11-s + 1.83·12-s + 4.15·13-s − 2.63·14-s + 1.83·15-s + 16-s − 4.34·17-s − 0.378·18-s − 1.11·19-s + 20-s + 4.84·21-s − 4.43·22-s + 8.02·23-s − 1.83·24-s + 25-s − 4.15·26-s − 4.81·27-s + 2.63·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.447·5-s − 0.750·6-s + 0.996·7-s − 0.353·8-s + 0.126·9-s − 0.316·10-s + 1.33·11-s + 0.530·12-s + 1.15·13-s − 0.704·14-s + 0.474·15-s + 0.250·16-s − 1.05·17-s − 0.0890·18-s − 0.254·19-s + 0.223·20-s + 1.05·21-s − 0.946·22-s + 1.67·23-s − 0.375·24-s + 0.200·25-s − 0.815·26-s − 0.927·27-s + 0.498·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.058357944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.058357944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 9.88T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 - 5.82T + 47T^{2} \) |
| 53 | \( 1 - 8.37T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 9.87T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 + 9.03T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 4.51T + 97T^{2} \) |
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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
−8.511861923735178189539999658349, −7.42364501608905690108032858033, −6.91358068669653692540829232966, −6.11041234007929813983393819691, −5.28718855740846119748990213003, −4.20547740571445374827249925312, −3.56111707886072104162960972140, −2.54880584428605539338516029970, −1.80017909960774077910955502221, −1.06108912076427510912998012204,
1.06108912076427510912998012204, 1.80017909960774077910955502221, 2.54880584428605539338516029970, 3.56111707886072104162960972140, 4.20547740571445374827249925312, 5.28718855740846119748990213003, 6.11041234007929813983393819691, 6.91358068669653692540829232966, 7.42364501608905690108032858033, 8.511861923735178189539999658349
