| L(s) = 1 | + 4.12e4i·2-s − 1.10e8i·3-s + 6.89e9·4-s + 4.56e12·6-s + 7.02e13i·7-s + 6.38e14i·8-s − 6.69e15·9-s − 5.33e16·11-s − 7.62e17i·12-s − 1.55e18i·13-s − 2.89e18·14-s + 3.28e19·16-s − 1.77e20i·17-s − 2.75e20i·18-s + 1.06e20·19-s + ⋯ |
| L(s) = 1 | + 0.444i·2-s − 1.48i·3-s + 0.802·4-s + 0.660·6-s + 0.799i·7-s + 0.801i·8-s − 1.20·9-s − 0.349·11-s − 1.19i·12-s − 0.649i·13-s − 0.355·14-s + 0.445·16-s − 0.886i·17-s − 0.535i·18-s + 0.0850·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(0.08886043252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08886043252\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 4.12e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 1.10e8iT - 5.55e15T^{2} \) |
| 7 | \( 1 - 7.02e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 + 5.33e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.55e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 1.77e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 - 1.06e20T + 1.58e42T^{2} \) |
| 23 | \( 1 - 5.14e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 + 8.16e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 3.64e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 9.53e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 2.96e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 5.30e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 - 6.47e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 1.72e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 + 3.23e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 5.51e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.40e30iT - 1.82e60T^{2} \) |
| 71 | \( 1 + 6.55e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 2.45e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 + 2.98e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 4.29e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 - 3.69e31T + 2.13e64T^{2} \) |
| 97 | \( 1 - 6.32e32iT - 3.65e65T^{2} \) |
| 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
−11.00603332197598935581598045902, −9.156716511175940184953116500660, −7.68194416298961337423865077972, −7.39782213698565974125083928023, −6.04652183205420121480037965457, −5.42947188652712963286457304900, −3.08030165325481622207351626077, −2.19896849336909042724912681967, −1.34599694360853468122689111844, −0.01439142336375043259397951609,
1.44100369333759670635089529057, 2.74363010512072620380235716382, 3.81085219509756151611701249111, 4.53515074592849015900588900635, 6.02227585568383233664474328047, 7.27897558570600634572312186670, 8.802024253649082591711685955519, 10.15628298102856709282332084304, 10.54863277721433276478162433540, 11.53714477790844684635298203048
