| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 16-s − 18-s − 24-s + 27-s − 2·31-s − 32-s + 36-s + 48-s − 49-s + 2·53-s − 54-s + 2·62-s + 64-s − 72-s − 2·79-s + 81-s − 2·83-s − 2·93-s − 96-s + 98-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 16-s − 18-s − 24-s + 27-s − 2·31-s − 32-s + 36-s + 48-s − 49-s + 2·53-s − 54-s + 2·62-s + 64-s − 72-s − 2·79-s + 81-s − 2·83-s − 2·93-s − 96-s + 98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8421207271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8421207271\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.60680338684477543005773904180, −9.823853253161526009705134188723, −9.065075104517322672543044717405, −8.402048725226824169854577677905, −7.49693022026998211474968996948, −6.84216139122036904630645940159, −5.54985663756684343507641843722, −3.95101372421565346112096838055, −2.83520846879413765808446134592, −1.67219952055531524557687854034,
1.67219952055531524557687854034, 2.83520846879413765808446134592, 3.95101372421565346112096838055, 5.54985663756684343507641843722, 6.84216139122036904630645940159, 7.49693022026998211474968996948, 8.402048725226824169854577677905, 9.065075104517322672543044717405, 9.823853253161526009705134188723, 10.60680338684477543005773904180
