| L(s) = 1 | − 2.08·3-s − 2.23·5-s + 1.92·7-s + 1.36·9-s − 4.71·11-s − 3.47·13-s + 4.67·15-s − 7.05·17-s − 7.10·19-s − 4.03·21-s + 4.99·23-s + 0.0172·25-s + 3.41·27-s − 1.87·29-s + 2.74·31-s + 9.86·33-s − 4.32·35-s − 0.747·37-s + 7.25·39-s − 11.3·41-s + 2.70·43-s − 3.05·45-s − 6.57·47-s − 3.27·49-s + 14.7·51-s − 8.29·53-s + 10.5·55-s + ⋯ |
| L(s) = 1 | − 1.20·3-s − 1.00·5-s + 0.729·7-s + 0.454·9-s − 1.42·11-s − 0.963·13-s + 1.20·15-s − 1.71·17-s − 1.63·19-s − 0.879·21-s + 1.04·23-s + 0.00344·25-s + 0.657·27-s − 0.347·29-s + 0.493·31-s + 1.71·33-s − 0.730·35-s − 0.122·37-s + 1.16·39-s − 1.76·41-s + 0.412·43-s − 0.455·45-s − 0.959·47-s − 0.468·49-s + 2.06·51-s − 1.13·53-s + 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09741409563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09741409563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.08T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - 1.92T + 7T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 + 7.05T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 + 0.747T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 + 6.57T + 47T^{2} \) |
| 53 | \( 1 + 8.29T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 + 2.65T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 6.62T + 83T^{2} \) |
| 89 | \( 1 - 0.801T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.238186822121915493756579116346, −7.79543013411063422310221933471, −6.83763049434015395602030427128, −6.37393954174480759526576016116, −5.10679646915219281405537708503, −4.91786796671589993398291484173, −4.20652217110008560619832270276, −2.88526855913258380330851921318, −1.93664284482130962236235132528, −0.17814961978883909875912471542,
0.17814961978883909875912471542, 1.93664284482130962236235132528, 2.88526855913258380330851921318, 4.20652217110008560619832270276, 4.91786796671589993398291484173, 5.10679646915219281405537708503, 6.37393954174480759526576016116, 6.83763049434015395602030427128, 7.79543013411063422310221933471, 8.238186822121915493756579116346
