| L(s) = 1 | − 1.30·2-s + 3-s − 0.302·4-s + 5-s − 1.30·6-s + 7-s + 3·8-s − 2·9-s − 1.30·10-s + 5.60·11-s − 0.302·12-s − 1.30·14-s + 15-s − 3.30·16-s + 0.394·17-s + 2.60·18-s − 1.60·19-s − 0.302·20-s + 21-s − 7.30·22-s − 3·23-s + 3·24-s + 25-s − 5·27-s − 0.302·28-s + 8.21·29-s − 1.30·30-s + ⋯ |
| L(s) = 1 | − 0.921·2-s + 0.577·3-s − 0.151·4-s + 0.447·5-s − 0.531·6-s + 0.377·7-s + 1.06·8-s − 0.666·9-s − 0.411·10-s + 1.69·11-s − 0.0874·12-s − 0.348·14-s + 0.258·15-s − 0.825·16-s + 0.0956·17-s + 0.614·18-s − 0.368·19-s − 0.0677·20-s + 0.218·21-s − 1.55·22-s − 0.625·23-s + 0.612·24-s + 0.200·25-s − 0.962·27-s − 0.0572·28-s + 1.52·29-s − 0.237·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243550320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243550320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 17 | \( 1 - 0.394T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 8.21T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 - 16.8T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 + 8.21T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−9.848981725181921041335486658978, −9.302800460910642422686796785502, −8.533947847500030824966407201022, −8.103228458667874422393690402771, −6.90231602901674378635086119089, −6.01308383548752237342626481131, −4.71271240788445316687481037336, −3.77097899680713502082254788357, −2.31251535246700753369112823984, −1.09368447304210276226495446749,
1.09368447304210276226495446749, 2.31251535246700753369112823984, 3.77097899680713502082254788357, 4.71271240788445316687481037336, 6.01308383548752237342626481131, 6.90231602901674378635086119089, 8.103228458667874422393690402771, 8.533947847500030824966407201022, 9.302800460910642422686796785502, 9.848981725181921041335486658978
