| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (−1.77 − 1.35i)5-s + (0.142 + 0.989i)6-s + (−0.316 + 0.172i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (1.47 − 1.67i)10-s + (4.35 + 3.77i)11-s + (−0.997 + 0.0713i)12-s + (0.383 − 0.701i)13-s + (−0.149 − 0.328i)14-s + (−2.02 − 0.947i)15-s + (0.959 + 0.281i)16-s + (0.369 + 0.276i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.795 − 0.606i)5-s + (0.0580 + 0.404i)6-s + (−0.119 + 0.0653i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (0.467 − 0.530i)10-s + (1.31 + 1.13i)11-s + (−0.287 + 0.0205i)12-s + (0.106 − 0.194i)13-s + (−0.0400 − 0.0877i)14-s + (−0.522 − 0.244i)15-s + (0.239 + 0.0704i)16-s + (0.0895 + 0.0670i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56236 + 0.321538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56236 + 0.321538i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (1.77 + 1.35i)T \) |
| 23 | \( 1 + (-4.25 + 2.21i)T \) |
| good | 7 | \( 1 + (0.316 - 0.172i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-4.35 - 3.77i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.383 + 0.701i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-0.369 - 0.276i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.891 + 6.20i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-9.47 + 1.36i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.31 + 0.845i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-8.32 - 3.10i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.67 - 3.66i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.53 + 7.03i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (1.40 + 1.40i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.849 - 1.55i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-4.03 - 13.7i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (4.25 + 6.61i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (11.7 + 0.842i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (2.46 + 2.84i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (9.68 + 12.9i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (3.46 - 1.01i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.268 - 0.718i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-3.65 - 2.35i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (0.336 + 0.902i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.27729186855070099089152670325, −9.218536817631624363413367319023, −8.868132647943103292909061815210, −7.87498032360343287862799120005, −7.08585604700808421723238500573, −6.37579616996938667390781037223, −4.75993048919625792629240954591, −4.34637935824298999397159142460, −2.97393506004543392490688493699, −1.09361948068593377581318645482,
1.21858576710043428146256218557, 2.96035182045153014231420421587, 3.58402164214175623996152686443, 4.45017049853689428937367337362, 6.01785436185252662689861828495, 6.97629504259812092313343809252, 8.103946938853964708650978371191, 8.668714499396243807492910921794, 9.639912158493089078786349659089, 10.43808105065449194361779642078
