L(s) = 1 | − 2.59·2-s + 3-s + 4.72·4-s − 2.59·6-s − 0.255·7-s − 7.06·8-s + 9-s + 5.72·11-s + 4.72·12-s − 0.592·13-s + 0.662·14-s + 8.86·16-s + 2.08·17-s − 2.59·18-s − 0.922·19-s − 0.255·21-s − 14.8·22-s + 5.19·23-s − 7.06·24-s + 1.53·26-s + 27-s − 1.20·28-s + 5·29-s + 2.05·31-s − 8.85·32-s + 5.72·33-s − 5.39·34-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 0.577·3-s + 2.36·4-s − 1.05·6-s − 0.0966·7-s − 2.49·8-s + 0.333·9-s + 1.72·11-s + 1.36·12-s − 0.164·13-s + 0.177·14-s + 2.21·16-s + 0.504·17-s − 0.611·18-s − 0.211·19-s − 0.0557·21-s − 3.16·22-s + 1.08·23-s − 1.44·24-s + 0.301·26-s + 0.192·27-s − 0.228·28-s + 0.928·29-s + 0.368·31-s − 1.56·32-s + 0.996·33-s − 0.925·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.186773159\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.186773159\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 7 | \( 1 + 0.255T + 7T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 0.592T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 + 0.922T + 19T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 2.05T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 - 1.27T + 41T^{2} \) |
| 43 | \( 1 - 9.95T + 43T^{2} \) |
| 53 | \( 1 - 3.57T + 53T^{2} \) |
| 59 | \( 1 - 0.151T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 3.41T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−8.912841868596746683901328217109, −7.959186658101122849206689846400, −7.35486390879776596481225382589, −6.63799547438037825717630367052, −6.11490331080120859838373806498, −4.67380939803969290220056052118, −3.54343412101246464047359541165, −2.70614106341405747399478472611, −1.63247060116833111382089343012, −0.886178972007289519101888924574,
0.886178972007289519101888924574, 1.63247060116833111382089343012, 2.70614106341405747399478472611, 3.54343412101246464047359541165, 4.67380939803969290220056052118, 6.11490331080120859838373806498, 6.63799547438037825717630367052, 7.35486390879776596481225382589, 7.959186658101122849206689846400, 8.912841868596746683901328217109