| L(s) = 1 | + 1.92·2-s + 1.68·4-s − 0.753·5-s + 7-s − 0.598·8-s − 1.44·10-s + 3.22·11-s − 4.98·13-s + 1.92·14-s − 4.52·16-s + 6.32·17-s − 2.95·19-s − 1.27·20-s + 6.19·22-s − 0.477·23-s − 4.43·25-s − 9.57·26-s + 1.68·28-s + 6.23·29-s + 9.65·31-s − 7.49·32-s + 12.1·34-s − 0.753·35-s − 0.00252·37-s − 5.67·38-s + 0.450·40-s + 7.44·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.844·4-s − 0.336·5-s + 0.377·7-s − 0.211·8-s − 0.457·10-s + 0.971·11-s − 1.38·13-s + 0.513·14-s − 1.13·16-s + 1.53·17-s − 0.678·19-s − 0.284·20-s + 1.31·22-s − 0.0995·23-s − 0.886·25-s − 1.87·26-s + 0.319·28-s + 1.15·29-s + 1.73·31-s − 1.32·32-s + 2.08·34-s − 0.127·35-s − 0.000415·37-s − 0.921·38-s + 0.0712·40-s + 1.16·41-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
| \(L(1)\) |
\(\approx\) |
\(3.990208060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.990208060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.92T + 2T^{2} \) |
| 5 | \( 1 + 0.753T + 5T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 + 4.98T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 2.95T + 19T^{2} \) |
| 23 | \( 1 + 0.477T + 23T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 + 0.00252T + 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 + 0.802T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 + 1.03T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 + 2.92T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 - 9.13T + 97T^{2} \) |
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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
−7.902769858027297179689895716096, −6.78772879525697382417766353062, −6.44511418046822364522533229276, −5.53295303397826581308170166708, −4.94355576674881894437423893519, −4.31637138205785490893935337036, −3.72321677164713333288800515680, −2.88599535194630940910345497977, −2.12299460707049853788351716969, −0.802143987705254648564888050110,
0.802143987705254648564888050110, 2.12299460707049853788351716969, 2.88599535194630940910345497977, 3.72321677164713333288800515680, 4.31637138205785490893935337036, 4.94355576674881894437423893519, 5.53295303397826581308170166708, 6.44511418046822364522533229276, 6.78772879525697382417766353062, 7.902769858027297179689895716096
