| L(s) = 1 | + 2.62·2-s − 3-s + 4.91·4-s − 0.281·5-s − 2.62·6-s − 7-s + 7.66·8-s + 9-s − 0.740·10-s + 5.88·11-s − 4.91·12-s + 0.529·13-s − 2.62·14-s + 0.281·15-s + 10.3·16-s − 4.53·17-s + 2.62·18-s + 6.62·19-s − 1.38·20-s + 21-s + 15.4·22-s + 6.22·23-s − 7.66·24-s − 4.92·25-s + 1.39·26-s − 27-s − 4.91·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.577·3-s + 2.45·4-s − 0.125·5-s − 1.07·6-s − 0.377·7-s + 2.70·8-s + 0.333·9-s − 0.234·10-s + 1.77·11-s − 1.41·12-s + 0.146·13-s − 0.702·14-s + 0.0726·15-s + 2.57·16-s − 1.09·17-s + 0.619·18-s + 1.52·19-s − 0.309·20-s + 0.218·21-s + 3.30·22-s + 1.29·23-s − 1.56·24-s − 0.984·25-s + 0.273·26-s − 0.192·27-s − 0.928·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.349471949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.349471949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 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 | \( 1 - 10.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.077293265203973838067473569587, −7.43868636068790262791110493554, −7.07837716483638441943552245361, −6.27233946577220405280728366564, −5.70369408217228265114878072134, −4.95254650212132728428926744062, −3.87308292195879454582471613098, −3.74214175015374695120262642257, −2.42159031347442722794772985690, −1.26450456552102927214022696568,
1.26450456552102927214022696568, 2.42159031347442722794772985690, 3.74214175015374695120262642257, 3.87308292195879454582471613098, 4.95254650212132728428926744062, 5.70369408217228265114878072134, 6.27233946577220405280728366564, 7.07837716483638441943552245361, 7.43868636068790262791110493554, 9.077293265203973838067473569587
