L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.382 − 0.923i)3-s + (0.866 − 0.499i)4-s + (−0.923 − 0.382i)5-s + (−0.608 − 0.793i)6-s + (−0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.991 − 0.130i)10-s + (−0.793 − 0.608i)12-s + (−0.198 + 0.739i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s − 1.58i·19-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.382 − 0.923i)3-s + (0.866 − 0.499i)4-s + (−0.923 − 0.382i)5-s + (−0.608 − 0.793i)6-s + (−0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.991 − 0.130i)10-s + (−0.793 − 0.608i)12-s + (−0.198 + 0.739i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s − 1.58i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.292012425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292012425\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.382 + 0.923i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (0.258 + 0.965i)T \) |
good | 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.198 - 0.739i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 1.58iT - T^{2} \) |
| 23 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.866 - 0.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.486195678362084363672275134058, −7.67063469223652338376660896715, −7.01700332336464507967488532001, −6.57781698629818903352866026563, −5.57154872906159748126125495343, −4.57706893638900656788169650905, −4.12089502263627336709185757058, −2.99706946189420479173195086208, −1.90829377538553414345068980932, −0.62152909016125040958655401274,
2.26564844783646740716516007853, 3.36622651796571535570317694847, 3.77254625780776775077523723958, 4.69831780315617811732580879198, 5.67478098869274803327653826884, 5.96887837893221355356629577539, 6.96688923605984407482723424395, 8.123870513491685580088024492210, 8.272660727341772697700225990997, 9.661683399824528287984583624484