| L(s) = 1 | + 2-s + 3.06·3-s + 4-s + 2.10·5-s + 3.06·6-s + 7-s + 8-s + 6.36·9-s + 2.10·10-s + 0.927·11-s + 3.06·12-s − 4.84·13-s + 14-s + 6.44·15-s + 16-s + 4.42·17-s + 6.36·18-s − 1.06·19-s + 2.10·20-s + 3.06·21-s + 0.927·22-s − 5.38·23-s + 3.06·24-s − 0.561·25-s − 4.84·26-s + 10.2·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.76·3-s + 0.5·4-s + 0.942·5-s + 1.24·6-s + 0.377·7-s + 0.353·8-s + 2.12·9-s + 0.666·10-s + 0.279·11-s + 0.883·12-s − 1.34·13-s + 0.267·14-s + 1.66·15-s + 0.250·16-s + 1.07·17-s + 1.49·18-s − 0.243·19-s + 0.471·20-s + 0.667·21-s + 0.197·22-s − 1.12·23-s + 0.624·24-s − 0.112·25-s − 0.950·26-s + 1.98·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.880843000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.880843000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 11 | \( 1 - 0.927T + 11T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 + 1.06T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 - 0.403T + 37T^{2} \) |
| 41 | \( 1 + 0.225T + 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 - 0.950T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 0.809T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 1.13T + 67T^{2} \) |
| 71 | \( 1 + 0.0848T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 + 4.18T + 79T^{2} \) |
| 83 | \( 1 + 3.72T + 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 + 0.945T + 97T^{2} \) |
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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
−8.057623904444798867308028432096, −7.48809766726851441043987936273, −6.73187010237811748052088251817, −5.91631749138396113722481589921, −5.02074983524375528721355907151, −4.34947738867233899048647975562, −3.52439425072101780768099663102, −2.71573957792899137195819172194, −2.17333255631656517238369393261, −1.42220218691223061460101560100,
1.42220218691223061460101560100, 2.17333255631656517238369393261, 2.71573957792899137195819172194, 3.52439425072101780768099663102, 4.34947738867233899048647975562, 5.02074983524375528721355907151, 5.91631749138396113722481589921, 6.73187010237811748052088251817, 7.48809766726851441043987936273, 8.057623904444798867308028432096
