| L(s) = 1 | − 2.42·2-s + 3.89·4-s − 4.13·5-s + 7-s − 4.59·8-s + 10.0·10-s + 4.39·11-s + 0.702·13-s − 2.42·14-s + 3.37·16-s + 0.979·17-s − 7.50·19-s − 16.0·20-s − 10.6·22-s + 4.28·23-s + 12.0·25-s − 1.70·26-s + 3.89·28-s − 8.97·29-s − 0.872·31-s + 0.999·32-s − 2.37·34-s − 4.13·35-s + 6.83·37-s + 18.2·38-s + 19.0·40-s + 4.36·41-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.94·4-s − 1.84·5-s + 0.377·7-s − 1.62·8-s + 3.17·10-s + 1.32·11-s + 0.194·13-s − 0.648·14-s + 0.844·16-s + 0.237·17-s − 1.72·19-s − 3.59·20-s − 2.27·22-s + 0.893·23-s + 2.41·25-s − 0.334·26-s + 0.735·28-s − 1.66·29-s − 0.156·31-s + 0.176·32-s − 0.407·34-s − 0.698·35-s + 1.12·37-s + 2.95·38-s + 3.00·40-s + 0.681·41-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
| \(L(1)\) |
\(\approx\) |
\(0.4930520818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4930520818\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.42T + 2T^{2} \) |
| 5 | \( 1 + 4.13T + 5T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 - 0.702T + 13T^{2} \) |
| 17 | \( 1 - 0.979T + 17T^{2} \) |
| 19 | \( 1 + 7.50T + 19T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 + 0.872T + 31T^{2} \) |
| 37 | \( 1 - 6.83T + 37T^{2} \) |
| 41 | \( 1 - 4.36T + 41T^{2} \) |
| 43 | \( 1 + 6.34T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 - 0.431T + 61T^{2} \) |
| 67 | \( 1 - 1.64T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
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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
−7.993534657111163238259634102266, −7.42290554091374974940808158270, −6.78644599856403443810075527611, −6.26223722809088365970088369672, −4.90149437382028277534519327961, −4.05054218395217134830491227236, −3.56637188906877718554929974723, −2.36542126004225007716351799134, −1.35516363324293683534596319636, −0.49603155541254197757705582616,
0.49603155541254197757705582616, 1.35516363324293683534596319636, 2.36542126004225007716351799134, 3.56637188906877718554929974723, 4.05054218395217134830491227236, 4.90149437382028277534519327961, 6.26223722809088365970088369672, 6.78644599856403443810075527611, 7.42290554091374974940808158270, 7.993534657111163238259634102266
