| L(s) = 1 | + 2.44·2-s + 3.99·4-s − 1.89·5-s − 0.969·7-s + 4.87·8-s − 4.64·10-s − 11-s − 1.38·13-s − 2.37·14-s + 3.95·16-s − 0.855·17-s + 5.83·19-s − 7.57·20-s − 2.44·22-s − 3.24·23-s − 1.40·25-s − 3.37·26-s − 3.87·28-s − 7.17·29-s − 3.46·31-s − 0.0706·32-s − 2.09·34-s + 1.83·35-s + 5.28·37-s + 14.2·38-s − 9.25·40-s − 6.09·41-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 1.99·4-s − 0.848·5-s − 0.366·7-s + 1.72·8-s − 1.46·10-s − 0.301·11-s − 0.382·13-s − 0.634·14-s + 0.989·16-s − 0.207·17-s + 1.33·19-s − 1.69·20-s − 0.521·22-s − 0.675·23-s − 0.280·25-s − 0.662·26-s − 0.731·28-s − 1.33·29-s − 0.621·31-s − 0.0124·32-s − 0.359·34-s + 0.310·35-s + 0.869·37-s + 2.31·38-s − 1.46·40-s − 0.951·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 + 1.89T + 5T^{2} \) |
| 7 | \( 1 + 0.969T + 7T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 + 0.855T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 5.28T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 1.07T + 43T^{2} \) |
| 47 | \( 1 + 6.02T + 47T^{2} \) |
| 53 | \( 1 - 2.06T + 53T^{2} \) |
| 59 | \( 1 + 5.72T + 59T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 7.88T + 71T^{2} \) |
| 73 | \( 1 - 0.611T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
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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
−7.61168757520857064629931989698, −6.84408031917001586668887380197, −6.12404642226433660758527994250, −5.38612304461074579103392806414, −4.83763726044003869848945745396, −3.94479320832438193470464918329, −3.50024787284781557164392554056, −2.74744100355010287501212101416, −1.74763134168502988188394485148, 0,
1.74763134168502988188394485148, 2.74744100355010287501212101416, 3.50024787284781557164392554056, 3.94479320832438193470464918329, 4.83763726044003869848945745396, 5.38612304461074579103392806414, 6.12404642226433660758527994250, 6.84408031917001586668887380197, 7.61168757520857064629931989698
