| L(s) = 1 | − 0.886·3-s − 0.640·5-s + 1.27·7-s − 2.21·9-s + 11-s + 0.262·13-s + 0.568·15-s − 4.04·17-s + 3.16·19-s − 1.13·21-s − 8.22·23-s − 4.58·25-s + 4.62·27-s + 7.92·29-s − 2.15·31-s − 0.886·33-s − 0.818·35-s + 10.1·37-s − 0.233·39-s − 1.74·41-s − 5.26·43-s + 1.41·45-s + 3.30·47-s − 5.36·49-s + 3.58·51-s + 6.13·53-s − 0.640·55-s + ⋯ |
| L(s) = 1 | − 0.512·3-s − 0.286·5-s + 0.482·7-s − 0.737·9-s + 0.301·11-s + 0.0728·13-s + 0.146·15-s − 0.981·17-s + 0.725·19-s − 0.247·21-s − 1.71·23-s − 0.917·25-s + 0.889·27-s + 1.47·29-s − 0.387·31-s − 0.154·33-s − 0.138·35-s + 1.67·37-s − 0.0373·39-s − 0.272·41-s − 0.802·43-s + 0.211·45-s + 0.482·47-s − 0.766·49-s + 0.502·51-s + 0.842·53-s − 0.0864·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155958035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155958035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 0.886T + 3T^{2} \) |
| 5 | \( 1 + 0.640T + 5T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 13 | \( 1 - 0.262T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 8.22T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 1.74T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 - 3.30T + 47T^{2} \) |
| 53 | \( 1 - 6.13T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 - 0.846T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 0.399T + 71T^{2} \) |
| 73 | \( 1 + 3.76T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 2.44T + 97T^{2} \) |
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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
−8.062575553971278648881349029097, −7.48760229701679151944777068587, −6.38739808229865833853045013663, −6.11381503061779963907278149135, −5.18529493237828742056485271419, −4.50294248168326132095830804215, −3.76324240295960261390806852081, −2.74251720692309132605949249406, −1.82970312750676328892657626121, −0.57035165141410437121986571941,
0.57035165141410437121986571941, 1.82970312750676328892657626121, 2.74251720692309132605949249406, 3.76324240295960261390806852081, 4.50294248168326132095830804215, 5.18529493237828742056485271419, 6.11381503061779963907278149135, 6.38739808229865833853045013663, 7.48760229701679151944777068587, 8.062575553971278648881349029097
