L(s) = 1 | + (1.16 − 0.799i)2-s + (1.51 − 0.835i)3-s + (0.721 − 1.86i)4-s + (0.0912 + 2.23i)5-s + (1.10 − 2.18i)6-s + 1.22i·7-s + (−0.649 − 2.75i)8-s + (1.60 − 2.53i)9-s + (1.89 + 2.53i)10-s + (−1.19 − 3.68i)11-s + (−0.463 − 3.43i)12-s + (−1.75 + 5.41i)13-s + (0.982 + 1.43i)14-s + (2.00 + 3.31i)15-s + (−2.95 − 2.69i)16-s + (0.726 − 0.999i)17-s + ⋯ |
L(s) = 1 | + (0.824 − 0.565i)2-s + (0.875 − 0.482i)3-s + (0.360 − 0.932i)4-s + (0.0408 + 0.999i)5-s + (0.449 − 0.893i)6-s + 0.464i·7-s + (−0.229 − 0.973i)8-s + (0.534 − 0.845i)9-s + (0.598 + 0.801i)10-s + (−0.361 − 1.11i)11-s + (−0.133 − 0.990i)12-s + (−0.487 + 1.50i)13-s + (0.262 + 0.383i)14-s + (0.517 + 0.855i)15-s + (−0.739 − 0.672i)16-s + (0.176 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21454 - 1.17814i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21454 - 1.17814i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 + 0.799i)T \) |
| 3 | \( 1 + (-1.51 + 0.835i)T \) |
| 5 | \( 1 + (-0.0912 - 2.23i)T \) |
good | 7 | \( 1 - 1.22iT - 7T^{2} \) |
| 11 | \( 1 + (1.19 + 3.68i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.75 - 5.41i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.726 + 0.999i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.675 - 0.930i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.01 - 3.13i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.27 + 4.50i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.00 - 8.27i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.51 + 4.66i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.19 - 0.711i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.85iT - 43T^{2} \) |
| 47 | \( 1 + (-7.46 + 5.42i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (4.27 + 5.87i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.19 - 6.76i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.50 - 4.62i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.52 + 10.3i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-1.92 + 1.39i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.789 - 2.42i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (2.96 + 4.08i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.96 - 4.33i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (11.9 - 3.89i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.45 + 3.96i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.65174076245936830325514107600, −10.93935648258867191439991993689, −9.737794928935729493819154733158, −8.964662924846679996502369253355, −7.50946727589471249120815123888, −6.64843068716668872668785583100, −5.64460920646736081435575330589, −3.96255034721550506672930025481, −2.98260737076213530756056811179, −1.97674501625503597195362770324,
2.37261651421053695687132406798, 3.81202936258744877005848376467, 4.74040357326877520316476935424, 5.55361459735588817444496018488, 7.33497087975861833527510090008, 7.86275557540546211658526548312, 8.871611473877676290832292337642, 9.906555824443718800160134027213, 10.90027698368520267094932737459, 12.55975261707827075723283057574