| L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 1.53·6-s + 0.347·7-s + 8-s + 1.34·9-s + 1.53·12-s − 1.87·13-s + 0.347·14-s + 16-s − 1.87·17-s + 1.34·18-s + 19-s + 0.532·21-s + 1.53·23-s + 1.53·24-s − 1.87·26-s + 0.532·27-s + 0.347·28-s − 1.87·29-s + 32-s − 1.87·34-s + 1.34·36-s − 37-s + 38-s − 2.87·39-s + ⋯ |
| L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 1.53·6-s + 0.347·7-s + 8-s + 1.34·9-s + 1.53·12-s − 1.87·13-s + 0.347·14-s + 16-s − 1.87·17-s + 1.34·18-s + 19-s + 0.532·21-s + 1.53·23-s + 1.53·24-s − 1.87·26-s + 0.532·27-s + 0.347·28-s − 1.87·29-s + 32-s − 1.87·34-s + 1.34·36-s − 37-s + 38-s − 2.87·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.946561758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.946561758\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.53T + T^{2} \) |
| 7 | \( 1 - 0.347T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.87T + T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 23 | \( 1 - 1.53T + T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.53T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| 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
−8.616862596064748904976021066347, −7.77873318250970140648635171981, −7.17152920915400361154831934643, −6.77885521607385788888835106452, −5.30413696606658519391343074171, −4.87313184259353025648862862607, −3.96768468813104163989317715687, −3.16519910098316355134791707938, −2.41338293578327042636646666491, −1.83520021970847513060548334122,
1.83520021970847513060548334122, 2.41338293578327042636646666491, 3.16519910098316355134791707938, 3.96768468813104163989317715687, 4.87313184259353025648862862607, 5.30413696606658519391343074171, 6.77885521607385788888835106452, 7.17152920915400361154831934643, 7.77873318250970140648635171981, 8.616862596064748904976021066347
