| L(s) = 1 | + 1.90·3-s + 1.03·5-s + 0.216·7-s + 0.622·9-s + 3.51·11-s + 3.04·13-s + 1.96·15-s + 17-s + 4.47·19-s + 0.411·21-s − 5.89·23-s − 3.93·25-s − 4.52·27-s + 0.947·29-s + 6.79·31-s + 6.69·33-s + 0.223·35-s + 0.950·37-s + 5.80·39-s + 11.2·41-s + 0.955·43-s + 0.642·45-s + 5.04·47-s − 6.95·49-s + 1.90·51-s + 3.63·53-s + 3.62·55-s + ⋯ |
| L(s) = 1 | + 1.09·3-s + 0.461·5-s + 0.0817·7-s + 0.207·9-s + 1.06·11-s + 0.845·13-s + 0.506·15-s + 0.242·17-s + 1.02·19-s + 0.0898·21-s − 1.23·23-s − 0.787·25-s − 0.870·27-s + 0.175·29-s + 1.22·31-s + 1.16·33-s + 0.0377·35-s + 0.156·37-s + 0.928·39-s + 1.76·41-s + 0.145·43-s + 0.0957·45-s + 0.735·47-s − 0.993·49-s + 0.266·51-s + 0.499·53-s + 0.488·55-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
| \(L(1)\) |
\(\approx\) |
\(3.551205105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.551205105\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.90T + 3T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 - 0.216T + 7T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 - 0.947T + 29T^{2} \) |
| 31 | \( 1 - 6.79T + 31T^{2} \) |
| 37 | \( 1 - 0.950T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 0.955T + 43T^{2} \) |
| 47 | \( 1 - 5.04T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 61 | \( 1 + 2.00T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 + 8.63T + 71T^{2} \) |
| 73 | \( 1 - 5.98T + 73T^{2} \) |
| 79 | \( 1 + 0.883T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.394099900713050589591284396048, −7.935442939782899646884287403446, −7.11324285392442863358260373193, −6.08001878396995975987844905243, −5.75267247859071959398502596684, −4.39329853816994613672867634379, −3.76139409894678143633026803025, −2.95780057806067026791888815346, −2.05045662962113950672363445158, −1.10885605775061519309618518409,
1.10885605775061519309618518409, 2.05045662962113950672363445158, 2.95780057806067026791888815346, 3.76139409894678143633026803025, 4.39329853816994613672867634379, 5.75267247859071959398502596684, 6.08001878396995975987844905243, 7.11324285392442863358260373193, 7.935442939782899646884287403446, 8.394099900713050589591284396048
