| L(s) = 1 | + 2-s − 3.23·3-s + 4-s + 1.61·5-s − 3.23·6-s + 2.85·7-s + 8-s + 7.47·9-s + 1.61·10-s + 2.85·11-s − 3.23·12-s + 1.23·13-s + 2.85·14-s − 5.23·15-s + 16-s − 7.70·17-s + 7.47·18-s − 1.85·19-s + 1.61·20-s − 9.23·21-s + 2.85·22-s + 6.09·23-s − 3.23·24-s − 2.38·25-s + 1.23·26-s − 14.4·27-s + 2.85·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s + 0.723·5-s − 1.32·6-s + 1.07·7-s + 0.353·8-s + 2.49·9-s + 0.511·10-s + 0.860·11-s − 0.934·12-s + 0.342·13-s + 0.762·14-s − 1.35·15-s + 0.250·16-s − 1.86·17-s + 1.76·18-s − 0.425·19-s + 0.361·20-s − 2.01·21-s + 0.608·22-s + 1.26·23-s − 0.660·24-s − 0.476·25-s + 0.242·26-s − 2.78·27-s + 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.587343014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.587343014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4007 | \( 1+O(T) \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 0.291T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 0.618T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6.85T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.39254679496680201076473480847, −6.81464225209008002508330518478, −6.25934494338249563622065640976, −5.77906856840904420163906966088, −5.07367197985640799798212931818, −4.39686132902433672769485397881, −4.13216134321851127178589350015, −2.46729845883542613944601893479, −1.63303017502553530349473868666, −0.847437092911885731410319836871,
0.847437092911885731410319836871, 1.63303017502553530349473868666, 2.46729845883542613944601893479, 4.13216134321851127178589350015, 4.39686132902433672769485397881, 5.07367197985640799798212931818, 5.77906856840904420163906966088, 6.25934494338249563622065640976, 6.81464225209008002508330518478, 7.39254679496680201076473480847
