L(s) = 1 | − 1.61i·2-s + (1.26 + 1.26i)3-s − 1.61·4-s + (2.03 − 2.03i)6-s + (0.221 − 0.221i)7-s + i·8-s + 2.17i·9-s + (−2.03 − 2.03i)12-s + (−0.357 − 0.357i)14-s + (0.309 − 0.951i)17-s + 3.52·18-s + 0.557·21-s + (−1.26 + 1.26i)24-s + i·25-s + (−1.48 + 1.48i)27-s + (−0.357 + 0.357i)28-s + ⋯ |
L(s) = 1 | − 1.61i·2-s + (1.26 + 1.26i)3-s − 1.61·4-s + (2.03 − 2.03i)6-s + (0.221 − 0.221i)7-s + i·8-s + 2.17i·9-s + (−2.03 − 2.03i)12-s + (−0.357 − 0.357i)14-s + (0.309 − 0.951i)17-s + 3.52·18-s + 0.557·21-s + (−1.26 + 1.26i)24-s + i·25-s + (−1.48 + 1.48i)27-s + (−0.357 + 0.357i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.346132447\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.346132447\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 1.61iT - T^{2} \) |
| 3 | \( 1 + (-1.26 - 1.26i)T + iT^{2} \) |
| 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (1.26 + 1.26i)T + iT^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.17iT - T^{2} \) |
| 59 | \( 1 + 1.61iT - T^{2} \) |
| 61 | \( 1 + (1.39 - 1.39i)T - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.642 + 0.642i)T + iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (0.642 - 0.642i)T - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
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
−10.35288368516807251489070915387, −9.542142770875046362003247528899, −9.167478530373659456992748633535, −8.279084303159274878092212217250, −7.24522557810320702214811280806, −5.24966075159384666072122074519, −4.46247090255594849826074991755, −3.56346144148815042117010944740, −2.95224292272376695219224146868, −1.83704402982401397509200382824,
1.74748537697137233320452185668, 3.11060227590902797202070711377, 4.44045143367586847739925641252, 5.77088800365067881061283809805, 6.53042044374655625098902584477, 7.21436611902793590952466696829, 8.051040635193276788544333920243, 8.436732208733820339703100159372, 9.118585044841358588413282848099, 10.23710192302465354025523798128