| L(s) = 1 | − 2-s + 4-s − 2.92·5-s − 3.28·7-s − 8-s + 2.92·10-s + 0.882·11-s − 2.93·13-s + 3.28·14-s + 16-s − 2.03·17-s − 2.83·19-s − 2.92·20-s − 0.882·22-s + 7.21·23-s + 3.57·25-s + 2.93·26-s − 3.28·28-s + 1.57·29-s + 3.45·31-s − 32-s + 2.03·34-s + 9.61·35-s + 0.228·37-s + 2.83·38-s + 2.92·40-s − 7.96·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.30·5-s − 1.24·7-s − 0.353·8-s + 0.925·10-s + 0.266·11-s − 0.813·13-s + 0.877·14-s + 0.250·16-s − 0.492·17-s − 0.649·19-s − 0.654·20-s − 0.188·22-s + 1.50·23-s + 0.714·25-s + 0.575·26-s − 0.620·28-s + 0.293·29-s + 0.619·31-s − 0.176·32-s + 0.348·34-s + 1.62·35-s + 0.0374·37-s + 0.459·38-s + 0.462·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 0.882T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 23 | \( 1 - 7.21T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 - 0.228T + 37T^{2} \) |
| 41 | \( 1 + 7.96T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 - 6.05T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 8.51T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 3.74T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.44614478284861389397022250760, −6.85367531559705116872042679740, −6.53958087428923831201662708195, −5.43218756843903359392035525016, −4.52736439121440817039554935562, −3.78433363166322606916855913847, −3.06531953198206994623791898456, −2.32954244838057150951036291343, −0.847744651741033609852767401171, 0,
0.847744651741033609852767401171, 2.32954244838057150951036291343, 3.06531953198206994623791898456, 3.78433363166322606916855913847, 4.52736439121440817039554935562, 5.43218756843903359392035525016, 6.53958087428923831201662708195, 6.85367531559705116872042679740, 7.44614478284861389397022250760
