L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.239 − 1.66i)5-s + (0.841 + 0.540i)6-s + (0.415 − 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.698 + 1.53i)10-s + (−0.797 − 0.234i)11-s + (−0.959 − 0.281i)12-s + (−0.142 + 0.989i)14-s + (−1.10 + 1.27i)15-s + (0.415 − 0.909i)16-s + (−1.10 − 0.708i)17-s + (−0.142 − 0.989i)18-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.239 − 1.66i)5-s + (0.841 + 0.540i)6-s + (0.415 − 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.698 + 1.53i)10-s + (−0.797 − 0.234i)11-s + (−0.959 − 0.281i)12-s + (−0.142 + 0.989i)14-s + (−1.10 + 1.27i)15-s + (0.415 − 0.909i)16-s + (−1.10 − 0.708i)17-s + (−0.142 − 0.989i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3063888103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3063888103\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (-0.841 + 0.540i)T \) |
good | 5 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 17 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 31 | \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 41 | \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 67 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.959 - 0.281i)T^{2} \) |
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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
−8.585273521979643967404599062121, −8.204148058873242180519104229991, −7.61715709694720684976370504977, −6.67211405611734770048249240635, −5.92377250542770101999453004885, −4.91795564379325398315504140736, −4.42252003814373452873758280764, −2.37632970647880518748999579152, −1.32206530390648277184001222492, −0.33738152501058983014955225539,
2.21054231236852728889579051565, 2.83750072718187478003982173563, 3.89404808494369502512867413428, 5.01553371040296764761514692043, 6.21462146942153637534089935082, 6.64802812651987049321923098491, 7.44828359231527312595809105167, 8.566985988684692042046450677527, 9.001458718717561287706675892403, 10.08109439884145726146739719059