| L(s) = 1 | − 0.432·2-s − 1.81·4-s + 4.09·5-s + 7-s + 1.64·8-s − 1.77·10-s − 2.19·11-s − 2.36·13-s − 0.432·14-s + 2.91·16-s + 2.91·17-s + 6.73·19-s − 7.43·20-s + 0.948·22-s − 4.56·23-s + 11.8·25-s + 1.02·26-s − 1.81·28-s + 4.17·29-s − 0.485·31-s − 4.55·32-s − 1.25·34-s + 4.09·35-s + 5.48·37-s − 2.90·38-s + 6.75·40-s − 10.5·41-s + ⋯ |
| L(s) = 1 | − 0.305·2-s − 0.906·4-s + 1.83·5-s + 0.377·7-s + 0.582·8-s − 0.560·10-s − 0.661·11-s − 0.656·13-s − 0.115·14-s + 0.728·16-s + 0.705·17-s + 1.54·19-s − 1.66·20-s + 0.202·22-s − 0.951·23-s + 2.36·25-s + 0.200·26-s − 0.342·28-s + 0.774·29-s − 0.0871·31-s − 0.805·32-s − 0.215·34-s + 0.693·35-s + 0.902·37-s − 0.471·38-s + 1.06·40-s − 1.65·41-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
| \(L(1)\) |
\(\approx\) |
\(2.189493092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189493092\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.432T + 2T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 + 4.56T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 + 0.485T + 31T^{2} \) |
| 37 | \( 1 - 5.48T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 6.67T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 - 1.66T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 + 0.610T + 73T^{2} \) |
| 79 | \( 1 + 7.36T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 7.36T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.88554526411390793766220996727, −7.29369148372728469456770857913, −6.32328683439029544524670309416, −5.46117250202940403195995429719, −5.29271424736132670507333528885, −4.55981225188016285173039167711, −3.38801338383382777575095993931, −2.52885053860601469340282193023, −1.67325969676634930592525304773, −0.816021382635084012554089575550,
0.816021382635084012554089575550, 1.67325969676634930592525304773, 2.52885053860601469340282193023, 3.38801338383382777575095993931, 4.55981225188016285173039167711, 5.29271424736132670507333528885, 5.46117250202940403195995429719, 6.32328683439029544524670309416, 7.29369148372728469456770857913, 7.88554526411390793766220996727
