| L(s) = 1 | − 2.79·2-s + 5.80·4-s + 1.63·5-s + 7-s − 10.6·8-s − 4.57·10-s + 6.47·11-s + 4.75·13-s − 2.79·14-s + 18.0·16-s − 0.645·17-s + 5.60·19-s + 9.49·20-s − 18.0·22-s + 0.0762·23-s − 2.32·25-s − 13.2·26-s + 5.80·28-s − 1.88·29-s + 2.27·31-s − 29.2·32-s + 1.80·34-s + 1.63·35-s − 5.95·37-s − 15.6·38-s − 17.3·40-s + 8.30·41-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.90·4-s + 0.732·5-s + 0.377·7-s − 3.75·8-s − 1.44·10-s + 1.95·11-s + 1.31·13-s − 0.746·14-s + 4.51·16-s − 0.156·17-s + 1.28·19-s + 2.12·20-s − 3.85·22-s + 0.0158·23-s − 0.464·25-s − 2.60·26-s + 1.09·28-s − 0.349·29-s + 0.408·31-s − 5.16·32-s + 0.309·34-s + 0.276·35-s − 0.978·37-s − 2.53·38-s − 2.74·40-s + 1.29·41-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
| \(L(1)\) |
\(\approx\) |
\(1.450594751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450594751\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.79T + 2T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 + 0.645T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 0.0762T + 23T^{2} \) |
| 29 | \( 1 + 1.88T + 29T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 + 5.95T + 37T^{2} \) |
| 41 | \( 1 - 8.30T + 41T^{2} \) |
| 43 | \( 1 - 0.409T + 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 + 4.31T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 4.61T + 89T^{2} \) |
| 97 | \( 1 + 8.08T + 97T^{2} \) |
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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
−8.086754384484608634745436542105, −7.17027115407796497467325034582, −6.74105290192718151895193434847, −6.01509137351068113922208965640, −5.57107754449854422523548683956, −3.98565297495991608840013700974, −3.25385065471554827099906507345, −2.13834391695643256382448506840, −1.41699001079377265647565537845, −0.930433836898779631668769715138,
0.930433836898779631668769715138, 1.41699001079377265647565537845, 2.13834391695643256382448506840, 3.25385065471554827099906507345, 3.98565297495991608840013700974, 5.57107754449854422523548683956, 6.01509137351068113922208965640, 6.74105290192718151895193434847, 7.17027115407796497467325034582, 8.086754384484608634745436542105
