L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.965 − 0.258i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + 1.00i·9-s + (0.258 + 0.965i)10-s + (−0.965 + 0.258i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s + (0.866 − 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s − 1.93·19-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.965 − 0.258i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + 1.00i·9-s + (0.258 + 0.965i)10-s + (−0.965 + 0.258i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s + (0.866 − 0.500i)15-s + (−0.5 − 0.866i)16-s + (0.500 + 0.866i)18-s − 1.93·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2408881817\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2408881817\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.258 - 0.965i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.93T + T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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
−8.517228421678219308034872794400, −7.56482901283269183499365694552, −6.77633563397311624196323023942, −6.27519191205863109700578369504, −5.69468685491649553648385377836, −4.66272709645156986651681482101, −3.75176971487235294423003177353, −2.51044359934675329292761107364, −2.25818993493534053808459622760, −0.11438479823835515555904042000,
2.11490976645207411789078956637, 3.56200073328625341287592870562, 4.22252169136758393987546105776, 4.67999110820410492459758956394, 5.62972920675763566575400026637, 6.30137072122235105227953208334, 7.01240295637409490474341187902, 7.889061958580918233618384233308, 8.807281631098795788773995980627, 9.544006245680370003465993155594