| L(s) = 1 | + 2.22·3-s − 1.27·5-s + 0.396·7-s + 1.93·9-s − 2.72·11-s + 6.91·13-s − 2.84·15-s + 17-s + 0.0994·19-s + 0.881·21-s + 6.04·23-s − 3.36·25-s − 2.36·27-s − 0.610·29-s − 6.40·31-s − 6.05·33-s − 0.507·35-s + 11.3·37-s + 15.3·39-s + 2.02·41-s + 9.63·43-s − 2.47·45-s + 13.2·47-s − 6.84·49-s + 2.22·51-s − 1.02·53-s + 3.48·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 0.571·5-s + 0.149·7-s + 0.644·9-s − 0.821·11-s + 1.91·13-s − 0.733·15-s + 0.242·17-s + 0.0228·19-s + 0.192·21-s + 1.26·23-s − 0.672·25-s − 0.455·27-s − 0.113·29-s − 1.14·31-s − 1.05·33-s − 0.0857·35-s + 1.86·37-s + 2.46·39-s + 0.316·41-s + 1.46·43-s − 0.368·45-s + 1.92·47-s − 0.977·49-s + 0.311·51-s − 0.141·53-s + 0.469·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.932443367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.932443367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 1.27T + 5T^{2} \) |
| 7 | \( 1 - 0.396T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 - 6.91T + 13T^{2} \) |
| 19 | \( 1 - 0.0994T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 + 0.610T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 - 9.63T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 1.02T + 53T^{2} \) |
| 61 | \( 1 - 0.211T + 61T^{2} \) |
| 67 | \( 1 + 8.77T + 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 + 6.43T + 73T^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 5.49T + 97T^{2} \) |
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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
−8.444655888513113974804060393235, −7.71462559548562867440671584714, −7.45895701698786943551717983219, −6.18961473100510738473761392406, −5.54966362691097948206750513501, −4.36432349843873336601730533126, −3.69636839843497880467164163208, −3.05196721837518073820052984179, −2.16361512784481781859036652725, −0.951815349928679019398384595291,
0.951815349928679019398384595291, 2.16361512784481781859036652725, 3.05196721837518073820052984179, 3.69636839843497880467164163208, 4.36432349843873336601730533126, 5.54966362691097948206750513501, 6.18961473100510738473761392406, 7.45895701698786943551717983219, 7.71462559548562867440671584714, 8.444655888513113974804060393235
