| L(s) = 1 | − 2.62·2-s + 4.91·4-s + 0.281·5-s − 7-s − 7.66·8-s − 0.740·10-s − 5.88·11-s + 0.529·13-s + 2.62·14-s + 10.3·16-s + 4.53·17-s + 6.62·19-s + 1.38·20-s + 15.4·22-s − 6.22·23-s − 4.92·25-s − 1.39·26-s − 4.91·28-s + 8.13·29-s − 6.74·31-s − 11.8·32-s − 11.9·34-s − 0.281·35-s + 4.51·37-s − 17.4·38-s − 2.15·40-s + 1.07·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.45·4-s + 0.125·5-s − 0.377·7-s − 2.70·8-s − 0.234·10-s − 1.77·11-s + 0.146·13-s + 0.702·14-s + 2.57·16-s + 1.09·17-s + 1.52·19-s + 0.309·20-s + 3.30·22-s − 1.29·23-s − 0.984·25-s − 0.273·26-s − 0.928·28-s + 1.51·29-s − 1.21·31-s − 2.08·32-s − 2.04·34-s − 0.0475·35-s + 0.743·37-s − 2.82·38-s − 0.340·40-s + 0.168·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 - 0.281T + 5T^{2} \) |
| 11 | \( 1 + 5.88T + 11T^{2} \) |
| 13 | \( 1 - 0.529T + 13T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 5.97T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 1.42T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 - 5.38T + 79T^{2} \) |
| 83 | \( 1 + 5.24T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 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
−7.66870892676611306212179687452, −7.35632907855141919190808873884, −6.05612105779823072420310720530, −5.89705001538190843696687995362, −4.86547673485568978903368595836, −3.43317424573209710697085880217, −2.80505824003724092344850755952, −2.01398286060592675017057529030, −0.986860515291410759584333709491, 0,
0.986860515291410759584333709491, 2.01398286060592675017057529030, 2.80505824003724092344850755952, 3.43317424573209710697085880217, 4.86547673485568978903368595836, 5.89705001538190843696687995362, 6.05612105779823072420310720530, 7.35632907855141919190808873884, 7.66870892676611306212179687452
