| L(s) = 1 | + (−0.900 − 0.433i)2-s + (−0.234 + 3.13i)3-s + (0.623 + 0.781i)4-s + (0.955 + 0.294i)5-s + (1.56 − 2.71i)6-s + (−2.53 − 4.38i)7-s + (−0.222 − 0.974i)8-s + (−6.78 − 1.02i)9-s + (−0.733 − 0.680i)10-s + (−2.91 + 3.65i)11-s + (−2.59 + 1.76i)12-s + (1.14 − 1.06i)13-s + (0.378 + 5.04i)14-s + (−1.14 + 2.92i)15-s + (−0.222 + 0.974i)16-s + (−3.80 + 1.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.306i)2-s + (−0.135 + 1.80i)3-s + (0.311 + 0.390i)4-s + (0.427 + 0.131i)5-s + (0.640 − 1.10i)6-s + (−0.956 − 1.65i)7-s + (−0.0786 − 0.344i)8-s + (−2.26 − 0.340i)9-s + (−0.231 − 0.215i)10-s + (−0.878 + 1.10i)11-s + (−0.748 + 0.510i)12-s + (0.318 − 0.295i)13-s + (0.101 + 1.34i)14-s + (−0.296 + 0.754i)15-s + (−0.0556 + 0.243i)16-s + (−0.922 + 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0411837 - 0.127540i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0411837 - 0.127540i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (-0.955 - 0.294i)T \) |
| 43 | \( 1 + (-0.631 + 6.52i)T \) |
| good | 3 | \( 1 + (0.234 - 3.13i)T + (-2.96 - 0.447i)T^{2} \) |
| 7 | \( 1 + (2.53 + 4.38i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.91 - 3.65i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.14 + 1.06i)T + (0.971 - 12.9i)T^{2} \) |
| 17 | \( 1 + (3.80 - 1.17i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (5.98 - 0.902i)T + (18.1 - 5.60i)T^{2} \) |
| 23 | \( 1 + (1.88 + 4.81i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.0458 - 0.612i)T + (-28.6 + 4.32i)T^{2} \) |
| 31 | \( 1 + (2.41 - 1.64i)T + (11.3 - 28.8i)T^{2} \) |
| 37 | \( 1 + (0.518 - 0.898i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.48 - 4.08i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-4.49 - 5.63i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.342 - 0.317i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.941 + 4.12i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-2.43 - 1.66i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (12.7 - 1.91i)T + (64.0 - 19.7i)T^{2} \) |
| 71 | \( 1 + (5.62 - 14.3i)T + (-52.0 - 48.2i)T^{2} \) |
| 73 | \( 1 + (-2.46 + 2.28i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (0.941 + 1.63i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.289 - 3.86i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (-0.485 + 6.47i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (9.71 - 12.1i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.81353560316897287956975152642, −10.68280936689299554525828267019, −10.08121675808787737589214638714, −9.404282174496972372843419924371, −8.391349166852973477915934637075, −7.09606427572049289944580247713, −6.06591497409248440694883637763, −4.50398557732012574466609567265, −3.97281366232197857261413216648, −2.65626495191934993744819153036,
0.095922139431005724173148923176, 2.02473044621198554783294085113, 2.75872619380047124797069000796, 5.63754689388544805770537117928, 6.00849942665925532719053550837, 6.76408837040192266847378045292, 7.924772231708357636845742063869, 8.747176133691891971018038145135, 9.214241477324984908352642034654, 10.79037134874976992400071927591
