| L(s) = 1 | + (−1.08 + 1.41i)2-s + (−0.543 + 0.0356i)3-s + (−0.309 − 1.15i)4-s + (−2.53 + 0.861i)5-s + (0.541 − 0.810i)6-s + (−1.32 − 0.549i)8-s + (−2.67 + 0.352i)9-s + (1.54 − 4.53i)10-s + (−2.21 − 4.49i)11-s + (0.209 + 0.618i)12-s + (−4.70 + 4.70i)13-s + (1.35 − 0.559i)15-s + (4.29 − 2.48i)16-s + (4.12 − 0.0371i)17-s + (2.41 − 4.18i)18-s + (4.21 + 3.23i)19-s + ⋯ |
| L(s) = 1 | + (−0.769 + 1.00i)2-s + (−0.314 + 0.0205i)3-s + (−0.154 − 0.578i)4-s + (−1.13 + 0.385i)5-s + (0.221 − 0.330i)6-s + (−0.468 − 0.194i)8-s + (−0.893 + 0.117i)9-s + (0.487 − 1.43i)10-s + (−0.668 − 1.35i)11-s + (0.0605 + 0.178i)12-s + (−1.30 + 1.30i)13-s + (0.348 − 0.144i)15-s + (1.07 − 0.620i)16-s + (0.999 − 0.00900i)17-s + (0.569 − 0.986i)18-s + (0.968 + 0.742i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.316793 + 0.0513921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.316793 + 0.0513921i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 + (-4.12 + 0.0371i)T \) |
| good | 2 | \( 1 + (1.08 - 1.41i)T + (-0.517 - 1.93i)T^{2} \) |
| 3 | \( 1 + (0.543 - 0.0356i)T + (2.97 - 0.391i)T^{2} \) |
| 5 | \( 1 + (2.53 - 0.861i)T + (3.96 - 3.04i)T^{2} \) |
| 11 | \( 1 + (2.21 + 4.49i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (4.70 - 4.70i)T - 13iT^{2} \) |
| 19 | \( 1 + (-4.21 - 3.23i)T + (4.91 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.267 - 4.08i)T + (-22.8 - 3.00i)T^{2} \) |
| 29 | \( 1 + (-1.90 + 9.55i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (0.0502 + 0.766i)T + (-30.7 + 4.04i)T^{2} \) |
| 37 | \( 1 + (5.32 + 2.62i)T + (22.5 + 29.3i)T^{2} \) |
| 41 | \( 1 + (-0.0510 - 0.256i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.397 + 0.959i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-8.29 - 2.22i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.81 - 0.238i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (3.63 + 4.73i)T + (-15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (0.999 + 1.13i)T + (-7.96 + 60.4i)T^{2} \) |
| 67 | \( 1 + (-10.7 - 6.18i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.95 - 2.64i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.852 + 0.747i)T + (9.52 + 72.3i)T^{2} \) |
| 79 | \( 1 + (-12.0 - 0.791i)T + (78.3 + 10.3i)T^{2} \) |
| 83 | \( 1 + (5.11 + 12.3i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (0.507 - 1.89i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (9.84 + 1.95i)T + (89.6 + 37.1i)T^{2} \) |
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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
−10.03010249032322954942600396318, −9.239538524120688921492694696671, −8.208220196872653951900619419509, −7.77585917935745858017057070145, −7.10732536719186310929018499970, −5.97411886507770755198121364887, −5.35229366298635405392262770442, −3.79509132630606423205034423910, −2.86024943597152844252361903686, −0.32207967376608911626817095676,
0.77473485067600468057611153170, 2.55450921097504422404842962283, 3.29982850424927638342853151947, 4.90984830985757280604858005949, 5.41610741932262071290126415204, 7.09404962476986576345415545007, 7.83009299902321164212626177564, 8.555945600946257881044234250665, 9.522574225836465984838270924019, 10.30863341776953445830575596328
