| L(s) = 1 | − 2-s − 1.87·5-s + 8-s + 9-s + 1.87·10-s + 0.347·11-s − 1.87·13-s − 16-s − 18-s − 0.347·22-s − 1.87·23-s + 2.53·25-s + 1.87·26-s + 1.53·31-s + 0.347·37-s − 1.87·40-s − 41-s − 1.87·45-s + 1.87·46-s + 49-s − 2.53·50-s + 1.53·53-s − 0.652·55-s + 0.347·59-s + 0.347·61-s − 1.53·62-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s − 1.87·5-s + 8-s + 9-s + 1.87·10-s + 0.347·11-s − 1.87·13-s − 16-s − 18-s − 0.347·22-s − 1.87·23-s + 2.53·25-s + 1.87·26-s + 1.53·31-s + 0.347·37-s − 1.87·40-s − 41-s − 1.87·45-s + 1.87·46-s + 49-s − 2.53·50-s + 1.53·53-s − 0.652·55-s + 0.347·59-s + 0.347·61-s − 1.53·62-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1999 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1999 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3681781978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3681781978\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1999 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.87T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.347T + T^{2} \) |
| 13 | \( 1 + 1.87T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.87T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 - 0.347T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 - 0.347T + T^{2} \) |
| 61 | \( 1 - 0.347T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.53T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 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
−9.374859577922917046390072306298, −8.342792734355486834706165437618, −7.919657173159473678276312491853, −7.30113857171750459893400983582, −6.73551704706833224973329780614, −5.00036671974030765248928438610, −4.34902647146543916775006278267, −3.77974589857458791413670290096, −2.25984212385598957636981139227, −0.68779805840383870749055707763,
0.68779805840383870749055707763, 2.25984212385598957636981139227, 3.77974589857458791413670290096, 4.34902647146543916775006278267, 5.00036671974030765248928438610, 6.73551704706833224973329780614, 7.30113857171750459893400983582, 7.919657173159473678276312491853, 8.342792734355486834706165437618, 9.374859577922917046390072306298
