| L(s) = 1 | − 2-s − 2.69·3-s + 4-s + 0.930·5-s + 2.69·6-s − 2.75·7-s − 8-s + 4.25·9-s − 0.930·10-s + 1.87·11-s − 2.69·12-s − 1.88·13-s + 2.75·14-s − 2.50·15-s + 16-s − 3.64·17-s − 4.25·18-s − 3.78·19-s + 0.930·20-s + 7.42·21-s − 1.87·22-s − 1.94·23-s + 2.69·24-s − 4.13·25-s + 1.88·26-s − 3.36·27-s − 2.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.55·3-s + 0.5·4-s + 0.416·5-s + 1.09·6-s − 1.04·7-s − 0.353·8-s + 1.41·9-s − 0.294·10-s + 0.566·11-s − 0.777·12-s − 0.523·13-s + 0.736·14-s − 0.647·15-s + 0.250·16-s − 0.883·17-s − 1.00·18-s − 0.868·19-s + 0.208·20-s + 1.62·21-s − 0.400·22-s − 0.405·23-s + 0.549·24-s − 0.826·25-s + 0.370·26-s − 0.648·27-s − 0.521·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07872431633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07872431633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 0.930T + 5T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 + 1.88T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 9.02T + 31T^{2} \) |
| 37 | \( 1 + 0.0607T + 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 - 9.48T + 43T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 + 5.02T + 59T^{2} \) |
| 61 | \( 1 + 7.43T + 61T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.49T + 73T^{2} \) |
| 79 | \( 1 + 5.20T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 3.11T + 97T^{2} \) |
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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
−7.48094036688901297296771695085, −7.12882758536428578513938274531, −6.28036019728499539607280790119, −6.01182257937732142867054031944, −5.36356757804465382073499285692, −4.32915313872706296389743886931, −3.63358957755078070956579334575, −2.33569102714986276906390054680, −1.57847507634257055929570410929, −0.16481075973508339009837902684,
0.16481075973508339009837902684, 1.57847507634257055929570410929, 2.33569102714986276906390054680, 3.63358957755078070956579334575, 4.32915313872706296389743886931, 5.36356757804465382073499285692, 6.01182257937732142867054031944, 6.28036019728499539607280790119, 7.12882758536428578513938274531, 7.48094036688901297296771695085
