| L(s) = 1 | − 2.16·2-s + 2.68·4-s + 2.65·5-s − 3.05·7-s − 1.47·8-s − 5.74·10-s + 4.20·11-s + 4.94·13-s + 6.60·14-s − 2.17·16-s − 0.493·17-s − 0.561·19-s + 7.12·20-s − 9.10·22-s + 3.83·23-s + 2.05·25-s − 10.7·26-s − 8.18·28-s − 6.08·29-s + 7.64·32-s + 1.06·34-s − 8.10·35-s + 5.57·37-s + 1.21·38-s − 3.92·40-s − 2.52·41-s − 10.3·43-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.34·4-s + 1.18·5-s − 1.15·7-s − 0.521·8-s − 1.81·10-s + 1.26·11-s + 1.37·13-s + 1.76·14-s − 0.542·16-s − 0.119·17-s − 0.128·19-s + 1.59·20-s − 1.94·22-s + 0.799·23-s + 0.410·25-s − 2.09·26-s − 1.54·28-s − 1.13·29-s + 1.35·32-s + 0.183·34-s − 1.37·35-s + 0.915·37-s + 0.197·38-s − 0.619·40-s − 0.394·41-s − 1.57·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208932494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208932494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 0.493T + 17T^{2} \) |
| 19 | \( 1 + 0.561T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 37 | \( 1 - 5.57T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 1.55T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 - 7.60T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 2.61T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 7.92T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.991743249352031873297306405309, −6.92482481978703163216562071224, −6.58523316160616492627831247726, −6.11542567342978969568447761440, −5.27436524539153160331187316703, −4.00662586679266633718969358061, −3.32990911007187166700417127624, −2.23678356009509821074131947851, −1.51893496298952864580394582362, −0.72401595265295331598208963212,
0.72401595265295331598208963212, 1.51893496298952864580394582362, 2.23678356009509821074131947851, 3.32990911007187166700417127624, 4.00662586679266633718969358061, 5.27436524539153160331187316703, 6.11542567342978969568447761440, 6.58523316160616492627831247726, 6.92482481978703163216562071224, 7.991743249352031873297306405309
