| L(s) = 1 | + 1.61·3-s + 2.61·5-s − 0.381·9-s + 4.47·11-s − 4.47·13-s + 4.23·15-s − 17-s + 6.47·19-s − 4.76·23-s + 1.85·25-s − 5.47·27-s + 2·29-s + 9.09·31-s + 7.23·33-s + 4·37-s − 7.23·39-s + 10.0·41-s − 10.0·43-s − 0.999·45-s + 11.2·47-s − 1.61·51-s − 3.09·53-s + 11.7·55-s + 10.4·57-s + 4·59-s − 7.61·61-s − 11.7·65-s + ⋯ |
| L(s) = 1 | + 0.934·3-s + 1.17·5-s − 0.127·9-s + 1.34·11-s − 1.24·13-s + 1.09·15-s − 0.242·17-s + 1.48·19-s − 0.993·23-s + 0.370·25-s − 1.05·27-s + 0.371·29-s + 1.63·31-s + 1.25·33-s + 0.657·37-s − 1.15·39-s + 1.57·41-s − 1.53·43-s − 0.149·45-s + 1.63·47-s − 0.226·51-s − 0.424·53-s + 1.57·55-s + 1.38·57-s + 0.520·59-s − 0.975·61-s − 1.45·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.797050162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.797050162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 3.09T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.941324683884433257266337829855, −7.44880120488527896452976758793, −6.47225672625500998585027008920, −6.01198927418271405001118508630, −5.15733433623260229622983960574, −4.33924656998569319803950923664, −3.43915054770371247339421615984, −2.59365546078979002359444578442, −2.06458656299424636767006347998, −0.989379091681628955044441504407,
0.989379091681628955044441504407, 2.06458656299424636767006347998, 2.59365546078979002359444578442, 3.43915054770371247339421615984, 4.33924656998569319803950923664, 5.15733433623260229622983960574, 6.01198927418271405001118508630, 6.47225672625500998585027008920, 7.44880120488527896452976758793, 7.941324683884433257266337829855
