| L(s) = 1 | − 1.28·5-s − 1.96·7-s + 2.40·11-s − 3.92·13-s + 3.63·17-s + 2.30·19-s + 4.55·23-s − 3.34·25-s + 5.81·29-s − 4.65·31-s + 2.52·35-s + 4.09·37-s − 7.34·41-s − 3.88·43-s − 7.51·47-s − 3.14·49-s + 10.6·53-s − 3.09·55-s − 10.8·59-s − 14.0·61-s + 5.04·65-s + 0.286·67-s + 10.1·71-s + 3.44·73-s − 4.73·77-s + 5.99·79-s − 10.0·83-s + ⋯ |
| L(s) = 1 | − 0.574·5-s − 0.742·7-s + 0.726·11-s − 1.08·13-s + 0.882·17-s + 0.529·19-s + 0.948·23-s − 0.669·25-s + 1.07·29-s − 0.836·31-s + 0.426·35-s + 0.672·37-s − 1.14·41-s − 0.592·43-s − 1.09·47-s − 0.448·49-s + 1.46·53-s − 0.417·55-s − 1.40·59-s − 1.79·61-s + 0.625·65-s + 0.0350·67-s + 1.20·71-s + 0.403·73-s − 0.539·77-s + 0.674·79-s − 1.10·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343644188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343644188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 - 4.55T + 23T^{2} \) |
| 29 | \( 1 - 5.81T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + 3.88T + 43T^{2} \) |
| 47 | \( 1 + 7.51T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 0.286T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 - 5.99T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 9.05T + 89T^{2} \) |
| 97 | \( 1 - 4.02T + 97T^{2} \) |
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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
−7.965026030272994598981312998934, −7.35928790568972555964528322418, −6.75671173174790749250653123079, −6.02940751747733801513910619524, −5.11734530071596428677925369015, −4.48508890222424990811994495222, −3.42126363551032204969124764996, −3.09376458736996657602643132728, −1.81165033501310444938528462250, −0.60523250832563126221853186895,
0.60523250832563126221853186895, 1.81165033501310444938528462250, 3.09376458736996657602643132728, 3.42126363551032204969124764996, 4.48508890222424990811994495222, 5.11734530071596428677925369015, 6.02940751747733801513910619524, 6.75671173174790749250653123079, 7.35928790568972555964528322418, 7.965026030272994598981312998934
