| L(s) = 1 | − 2.35·2-s + 3.55·4-s − 2.88·5-s − 7-s − 3.66·8-s + 6.79·10-s + 2.09·11-s + 0.909·13-s + 2.35·14-s + 1.51·16-s − 3.78·17-s + 7.58·19-s − 10.2·20-s − 4.92·22-s + 5.96·23-s + 3.30·25-s − 2.14·26-s − 3.55·28-s + 2.95·29-s − 0.925·31-s + 3.74·32-s + 8.92·34-s + 2.88·35-s − 0.864·37-s − 17.8·38-s + 10.5·40-s + 3.36·41-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.77·4-s − 1.28·5-s − 0.377·7-s − 1.29·8-s + 2.14·10-s + 0.630·11-s + 0.252·13-s + 0.629·14-s + 0.379·16-s − 0.918·17-s + 1.73·19-s − 2.28·20-s − 1.05·22-s + 1.24·23-s + 0.660·25-s − 0.420·26-s − 0.671·28-s + 0.548·29-s − 0.166·31-s + 0.661·32-s + 1.52·34-s + 0.487·35-s − 0.142·37-s − 2.89·38-s + 1.66·40-s + 0.525·41-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
| \(L(1)\) |
\(\approx\) |
\(0.6290051588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6290051588\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.35T + 2T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 - 0.909T + 13T^{2} \) |
| 17 | \( 1 + 3.78T + 17T^{2} \) |
| 19 | \( 1 - 7.58T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 + 0.925T + 31T^{2} \) |
| 37 | \( 1 + 0.864T + 37T^{2} \) |
| 41 | \( 1 - 3.36T + 41T^{2} \) |
| 43 | \( 1 - 7.26T + 43T^{2} \) |
| 47 | \( 1 + 1.13T + 47T^{2} \) |
| 53 | \( 1 - 0.941T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 - 2.67T + 67T^{2} \) |
| 71 | \( 1 + 3.06T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 - 9.71T + 83T^{2} \) |
| 89 | \( 1 + 0.340T + 89T^{2} \) |
| 97 | \( 1 - 8.16T + 97T^{2} \) |
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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
−7.77940796879501699382725006717, −7.39100332148714772609520861372, −6.87159314465596665701450874938, −6.13818597316258719661068081618, −5.01496546733689218939916010575, −4.12933757304175571877286197986, −3.33697438485626410044651041367, −2.53913540803195117725710099057, −1.26599008885749193927498486531, −0.58069169963719039884993237432,
0.58069169963719039884993237432, 1.26599008885749193927498486531, 2.53913540803195117725710099057, 3.33697438485626410044651041367, 4.12933757304175571877286197986, 5.01496546733689218939916010575, 6.13818597316258719661068081618, 6.87159314465596665701450874938, 7.39100332148714772609520861372, 7.77940796879501699382725006717
