| L(s) = 1 | − 0.103·3-s − 3.71·5-s − 2.98·9-s − 1.25·11-s + 0.166·13-s + 0.385·15-s + 5.79·17-s − 1.05·19-s − 8.98·23-s + 8.79·25-s + 0.621·27-s − 3.21·29-s − 9.67·31-s + 0.130·33-s + 1.70·37-s − 0.0173·39-s − 41-s − 3.83·43-s + 11.1·45-s − 7.32·47-s − 0.601·51-s + 2.82·53-s + 4.67·55-s + 0.109·57-s − 9.74·59-s − 4.82·61-s − 0.619·65-s + ⋯ |
| L(s) = 1 | − 0.0598·3-s − 1.66·5-s − 0.996·9-s − 0.379·11-s + 0.0462·13-s + 0.0995·15-s + 1.40·17-s − 0.242·19-s − 1.87·23-s + 1.75·25-s + 0.119·27-s − 0.596·29-s − 1.73·31-s + 0.0227·33-s + 0.280·37-s − 0.00277·39-s − 0.156·41-s − 0.584·43-s + 1.65·45-s − 1.06·47-s − 0.0842·51-s + 0.387·53-s + 0.630·55-s + 0.0145·57-s − 1.26·59-s − 0.617·61-s − 0.0768·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3844838218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3844838218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.103T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 11 | \( 1 + 1.25T + 11T^{2} \) |
| 13 | \( 1 - 0.166T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 23 | \( 1 + 8.98T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 9.67T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 43 | \( 1 + 3.83T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 9.74T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + 1.21T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 6.00T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 4.56T + 97T^{2} \) |
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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
−7.82116317769428654909205895295, −7.47279676762171337653701911886, −6.43774680246500113308280167698, −5.68980481632751172641072901020, −5.07241445704573518653685255555, −4.09251902094732645452374306934, −3.56217224113534847531331039452, −2.93674864756497881827373397853, −1.73449259948734618070751046195, −0.29762947337552829918190691215,
0.29762947337552829918190691215, 1.73449259948734618070751046195, 2.93674864756497881827373397853, 3.56217224113534847531331039452, 4.09251902094732645452374306934, 5.07241445704573518653685255555, 5.68980481632751172641072901020, 6.43774680246500113308280167698, 7.47279676762171337653701911886, 7.82116317769428654909205895295
