| L(s) = 1 | + 1.75·2-s + 1.09·4-s − 1.44·5-s − 7-s − 1.59·8-s − 2.53·10-s − 4.64·11-s − 3.75·13-s − 1.75·14-s − 4.99·16-s − 2.76·17-s + 1.65·19-s − 1.57·20-s − 8.16·22-s + 6.34·23-s − 2.91·25-s − 6.60·26-s − 1.09·28-s − 0.687·29-s − 4.29·31-s − 5.58·32-s − 4.85·34-s + 1.44·35-s − 1.53·37-s + 2.91·38-s + 2.30·40-s + 9.88·41-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.545·4-s − 0.645·5-s − 0.377·7-s − 0.564·8-s − 0.802·10-s − 1.39·11-s − 1.04·13-s − 0.469·14-s − 1.24·16-s − 0.670·17-s + 0.380·19-s − 0.352·20-s − 1.73·22-s + 1.32·23-s − 0.583·25-s − 1.29·26-s − 0.206·28-s − 0.127·29-s − 0.772·31-s − 0.986·32-s − 0.833·34-s + 0.244·35-s − 0.251·37-s + 0.473·38-s + 0.364·40-s + 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.531855017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.531855017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 - 6.34T + 23T^{2} \) |
| 29 | \( 1 + 0.687T + 29T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 + 1.53T + 37T^{2} \) |
| 41 | \( 1 - 9.88T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 - 0.153T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 - 0.908T + 59T^{2} \) |
| 61 | \( 1 - 8.22T + 61T^{2} \) |
| 67 | \( 1 + 9.93T + 67T^{2} \) |
| 71 | \( 1 - 8.33T + 71T^{2} \) |
| 73 | \( 1 - 8.16T + 73T^{2} \) |
| 79 | \( 1 + 1.51T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 1.44T + 89T^{2} \) |
| 97 | \( 1 + 9.30T + 97T^{2} \) |
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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
−7.49076781993489594041060522146, −7.22903516497108676843192913464, −6.28178273544463216110897848114, −5.44337019225728926597873857368, −5.07988359566806791119427570897, −4.30517686367011706290974094567, −3.65747069222636381770625084243, −2.77401854340988814251284680519, −2.34696020263904466825755776346, −0.47034786938284779619544578725,
0.47034786938284779619544578725, 2.34696020263904466825755776346, 2.77401854340988814251284680519, 3.65747069222636381770625084243, 4.30517686367011706290974094567, 5.07988359566806791119427570897, 5.44337019225728926597873857368, 6.28178273544463216110897848114, 7.22903516497108676843192913464, 7.49076781993489594041060522146
