| L(s) = 1 | − 2-s + 0.482·3-s + 4-s + 2.96·5-s − 0.482·6-s + 7-s − 8-s − 2.76·9-s − 2.96·10-s + 2.13·11-s + 0.482·12-s + 3.23·13-s − 14-s + 1.43·15-s + 16-s + 0.239·17-s + 2.76·18-s − 1.99·19-s + 2.96·20-s + 0.482·21-s − 2.13·22-s − 0.0927·23-s − 0.482·24-s + 3.80·25-s − 3.23·26-s − 2.78·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.278·3-s + 0.5·4-s + 1.32·5-s − 0.196·6-s + 0.377·7-s − 0.353·8-s − 0.922·9-s − 0.938·10-s + 0.645·11-s + 0.139·12-s + 0.896·13-s − 0.267·14-s + 0.369·15-s + 0.250·16-s + 0.0579·17-s + 0.652·18-s − 0.456·19-s + 0.663·20-s + 0.105·21-s − 0.456·22-s − 0.0193·23-s − 0.0984·24-s + 0.760·25-s − 0.633·26-s − 0.535·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.270927844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.270927844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 0.482T + 3T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 11 | \( 1 - 2.13T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 0.239T + 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 + 0.0927T + 23T^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 2.61T + 37T^{2} \) |
| 41 | \( 1 + 8.16T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 - 9.39T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 - 0.681T + 71T^{2} \) |
| 73 | \( 1 + 0.776T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 - 0.462T + 83T^{2} \) |
| 89 | \( 1 + 8.66T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
| 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
−8.323213467883131194540630438099, −7.47594985346695440755841428348, −6.55945006436634801568777667988, −6.01381072875082730179016426502, −5.54213130868834064516382666583, −4.44762590323922105418824598835, −3.40955983733357627931577332206, −2.51428155236866598328721742575, −1.82865623020281445778710667619, −0.900681025216271876442090512279,
0.900681025216271876442090512279, 1.82865623020281445778710667619, 2.51428155236866598328721742575, 3.40955983733357627931577332206, 4.44762590323922105418824598835, 5.54213130868834064516382666583, 6.01381072875082730179016426502, 6.55945006436634801568777667988, 7.47594985346695440755841428348, 8.323213467883131194540630438099
