| L(s) = 1 | − 0.860·5-s − 0.617·7-s − 4.88·11-s − 4.79·13-s − 2.56·17-s + 0.574·19-s + 1.04·23-s − 4.25·25-s + 3.36·29-s − 2.95·31-s + 0.531·35-s − 6.11·37-s + 9.79·41-s − 5.38·43-s + 6.46·47-s − 6.61·49-s + 2.44·53-s + 4.20·55-s − 1.74·59-s + 10.6·61-s + 4.12·65-s + 7.38·67-s − 7.40·71-s + 11.5·73-s + 3.01·77-s − 0.281·79-s + 8.62·83-s + ⋯ |
| L(s) = 1 | − 0.384·5-s − 0.233·7-s − 1.47·11-s − 1.33·13-s − 0.622·17-s + 0.131·19-s + 0.217·23-s − 0.851·25-s + 0.625·29-s − 0.530·31-s + 0.0897·35-s − 1.00·37-s + 1.53·41-s − 0.820·43-s + 0.942·47-s − 0.945·49-s + 0.336·53-s + 0.566·55-s − 0.227·59-s + 1.36·61-s + 0.511·65-s + 0.901·67-s − 0.879·71-s + 1.35·73-s + 0.343·77-s − 0.0317·79-s + 0.946·83-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
| \(L(1)\) |
\(\approx\) |
\(0.8088823976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8088823976\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.860T + 5T^{2} \) |
| 7 | \( 1 + 0.617T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 0.574T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 - 3.36T + 29T^{2} \) |
| 31 | \( 1 + 2.95T + 31T^{2} \) |
| 37 | \( 1 + 6.11T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 6.46T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 1.74T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 0.281T + 79T^{2} \) |
| 83 | \( 1 - 8.62T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 9.14T + 97T^{2} \) |
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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
−8.010541660827216543693937623936, −7.37886175535506034827598682523, −6.86938098534179186405434409947, −5.83561089847666510313785001816, −5.15328190914835464796035560532, −4.56127567616142715917259379758, −3.60730742227246135438144590354, −2.69134254927884663396108474501, −2.08649468429847892157791421104, −0.44140696810694134630459064345,
0.44140696810694134630459064345, 2.08649468429847892157791421104, 2.69134254927884663396108474501, 3.60730742227246135438144590354, 4.56127567616142715917259379758, 5.15328190914835464796035560532, 5.83561089847666510313785001816, 6.86938098534179186405434409947, 7.37886175535506034827598682523, 8.010541660827216543693937623936
