| L(s) = 1 | − 2.50·2-s − 3-s + 4.26·4-s + 5-s + 2.50·6-s − 5.67·8-s + 9-s − 2.50·10-s − 11-s − 4.26·12-s − 0.539·13-s − 15-s + 5.68·16-s + 7.89·17-s − 2.50·18-s + 3.19·19-s + 4.26·20-s + 2.50·22-s + 2.63·23-s + 5.67·24-s + 25-s + 1.35·26-s − 27-s + 8.97·29-s + 2.50·30-s + 10.4·31-s − 2.86·32-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.13·4-s + 0.447·5-s + 1.02·6-s − 2.00·8-s + 0.333·9-s − 0.791·10-s − 0.301·11-s − 1.23·12-s − 0.149·13-s − 0.258·15-s + 1.42·16-s + 1.91·17-s − 0.590·18-s + 0.733·19-s + 0.954·20-s + 0.533·22-s + 0.549·23-s + 1.15·24-s + 0.200·25-s + 0.264·26-s − 0.192·27-s + 1.66·29-s + 0.457·30-s + 1.87·31-s − 0.507·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040516934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040516934\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 13 | \( 1 + 0.539T + 13T^{2} \) |
| 17 | \( 1 - 7.89T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 - 8.97T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 3.69T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 0.364T + 61T^{2} \) |
| 67 | \( 1 + 2.15T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 8.19T + 97T^{2} \) |
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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
−7.87475098664223056477872251518, −7.35089792887813358374008630973, −6.65460746843149452562497107485, −5.93861058104482749914112680178, −5.35453857968793807657548300008, −4.37578341857322034379653467323, −3.00582501235995040579202175910, −2.48855420916749565539877706032, −1.09335622962025566307064475576, −0.902163644757244294208912312215,
0.902163644757244294208912312215, 1.09335622962025566307064475576, 2.48855420916749565539877706032, 3.00582501235995040579202175910, 4.37578341857322034379653467323, 5.35453857968793807657548300008, 5.93861058104482749914112680178, 6.65460746843149452562497107485, 7.35089792887813358374008630973, 7.87475098664223056477872251518
