| L(s) = 1 | − 0.154·2-s − 1.20·3-s − 1.97·4-s − 5-s + 0.186·6-s − 4.71·7-s + 0.615·8-s − 1.54·9-s + 0.154·10-s − 11-s + 2.38·12-s − 3.33·13-s + 0.730·14-s + 1.20·15-s + 3.85·16-s + 17-s + 0.239·18-s − 7.90·19-s + 1.97·20-s + 5.68·21-s + 0.154·22-s + 8.30·23-s − 0.742·24-s + 25-s + 0.516·26-s + 5.48·27-s + 9.32·28-s + ⋯ |
| L(s) = 1 | − 0.109·2-s − 0.695·3-s − 0.988·4-s − 0.447·5-s + 0.0762·6-s − 1.78·7-s + 0.217·8-s − 0.515·9-s + 0.0489·10-s − 0.301·11-s + 0.687·12-s − 0.925·13-s + 0.195·14-s + 0.311·15-s + 0.964·16-s + 0.242·17-s + 0.0564·18-s − 1.81·19-s + 0.441·20-s + 1.24·21-s + 0.0330·22-s + 1.73·23-s − 0.151·24-s + 0.200·25-s + 0.101·26-s + 1.05·27-s + 1.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2106425532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2106425532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 0.154T + 2T^{2} \) |
| 3 | \( 1 + 1.20T + 3T^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 19 | \( 1 + 7.90T + 19T^{2} \) |
| 23 | \( 1 - 8.30T + 23T^{2} \) |
| 29 | \( 1 + 3.44T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 5.64T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 9.59T + 53T^{2} \) |
| 59 | \( 1 + 1.53T + 59T^{2} \) |
| 61 | \( 1 + 0.388T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 6.22T + 71T^{2} \) |
| 73 | \( 1 - 9.76T + 73T^{2} \) |
| 79 | \( 1 + 2.96T + 79T^{2} \) |
| 83 | \( 1 + 6.61T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
| 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
−10.24326371331030912347393878837, −9.058497613990076328200960998158, −8.751598013333291097589672281452, −7.40608195618093037922038463013, −6.61062286346913631890862452861, −5.67486640321816118059612992592, −4.85494984287503363887112576783, −3.76128584106008457908200293408, −2.80666542199989542183495885164, −0.35871628932816571917236388406,
0.35871628932816571917236388406, 2.80666542199989542183495885164, 3.76128584106008457908200293408, 4.85494984287503363887112576783, 5.67486640321816118059612992592, 6.61062286346913631890862452861, 7.40608195618093037922038463013, 8.751598013333291097589672281452, 9.058497613990076328200960998158, 10.24326371331030912347393878837
