| L(s) = 1 | − 1.49·2-s + 0.220·4-s − 2.04·5-s + 7-s + 2.65·8-s + 3.05·10-s + 1.77·11-s − 4.05·13-s − 1.49·14-s − 4.39·16-s − 7.64·17-s + 3.24·19-s − 0.452·20-s − 2.64·22-s − 6.77·23-s − 0.807·25-s + 6.04·26-s + 0.220·28-s − 6.90·29-s − 1.66·31-s + 1.24·32-s + 11.3·34-s − 2.04·35-s + 9.03·37-s − 4.84·38-s − 5.42·40-s − 9.40·41-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.110·4-s − 0.915·5-s + 0.377·7-s + 0.937·8-s + 0.964·10-s + 0.535·11-s − 1.12·13-s − 0.398·14-s − 1.09·16-s − 1.85·17-s + 0.745·19-s − 0.101·20-s − 0.564·22-s − 1.41·23-s − 0.161·25-s + 1.18·26-s + 0.0417·28-s − 1.28·29-s − 0.298·31-s + 0.219·32-s + 1.95·34-s − 0.346·35-s + 1.48·37-s − 0.785·38-s − 0.858·40-s − 1.46·41-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
| \(L(1)\) |
\(\approx\) |
\(0.2697651382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2697651382\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.49T + 2T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + 7.64T + 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 - 9.03T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 - 1.14T + 47T^{2} \) |
| 53 | \( 1 + 0.909T + 53T^{2} \) |
| 59 | \( 1 - 0.510T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 0.730T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 - 3.42T + 73T^{2} \) |
| 79 | \( 1 + 7.04T + 79T^{2} \) |
| 83 | \( 1 + 1.32T + 83T^{2} \) |
| 89 | \( 1 - 3.63T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−7.76822740519952951299905248464, −7.50185682805880376645243333934, −6.78849767982248221677491529182, −5.84989499749629417424371868286, −4.77775459682507068345712285054, −4.35277971855172320747583361835, −3.65294995843675811061864788957, −2.34775380951657980026163975212, −1.63772333162806729280895591262, −0.30111629331361084461420006685,
0.30111629331361084461420006685, 1.63772333162806729280895591262, 2.34775380951657980026163975212, 3.65294995843675811061864788957, 4.35277971855172320747583361835, 4.77775459682507068345712285054, 5.84989499749629417424371868286, 6.78849767982248221677491529182, 7.50185682805880376645243333934, 7.76822740519952951299905248464
