| L(s) = 1 | + (0.0234 − 0.0875i)2-s + (2.37 + 1.83i)3-s + (3.45 + 1.99i)4-s + (3.59 − 3.47i)5-s + (0.216 − 0.164i)6-s + (−6.65 − 2.18i)7-s + (0.512 − 0.512i)8-s + (2.26 + 8.71i)9-s + (−0.220 − 0.396i)10-s + (−6.21 − 3.58i)11-s + (4.53 + 11.0i)12-s + (−2.42 + 2.42i)13-s + (−0.346 + 0.531i)14-s + (14.9 − 1.65i)15-s + (7.95 + 13.7i)16-s + (−4.60 − 17.1i)17-s + ⋯ |
| L(s) = 1 | + (0.0117 − 0.0437i)2-s + (0.790 + 0.611i)3-s + (0.864 + 0.498i)4-s + (0.718 − 0.695i)5-s + (0.0360 − 0.0274i)6-s + (−0.950 − 0.311i)7-s + (0.0640 − 0.0640i)8-s + (0.251 + 0.967i)9-s + (−0.0220 − 0.0396i)10-s + (−0.565 − 0.326i)11-s + (0.378 + 0.923i)12-s + (−0.186 + 0.186i)13-s + (−0.0247 + 0.0379i)14-s + (0.993 − 0.110i)15-s + (0.496 + 0.860i)16-s + (−0.270 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.93178 + 0.405622i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93178 + 0.405622i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.37 - 1.83i)T \) |
| 5 | \( 1 + (-3.59 + 3.47i)T \) |
| 7 | \( 1 + (6.65 + 2.18i)T \) |
| good | 2 | \( 1 + (-0.0234 + 0.0875i)T + (-3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (6.21 + 3.58i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (2.42 - 2.42i)T - 169iT^{2} \) |
| 17 | \( 1 + (4.60 + 17.1i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-6.28 - 10.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.355 - 0.0951i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 51.7T + 841T^{2} \) |
| 31 | \( 1 + (-2.02 - 1.16i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-17.3 - 4.65i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 44.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (45.6 + 45.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (54.7 + 14.6i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (17.1 + 63.8i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-65.6 - 37.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-93.3 + 53.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-8.01 - 29.9i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 69.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-7.00 - 26.1i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (68.4 - 39.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-77.2 - 77.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-18.4 + 10.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-45.5 - 45.5i)T + 9.40e3iT^{2} \) |
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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
−13.38993858490941323753414924632, −12.88027069121403816486923832183, −11.43193365576161332381049544851, −10.15884171490902971382601534557, −9.409274554326130575303118579258, −8.187527028684851205078354373085, −6.98099182865388778525804240065, −5.42756847764605410093358306869, −3.67998996637761318360383027103, −2.35616303715104506303494665185,
2.05091379468228111986994049281, 3.11861670517286527635188638879, 5.82410464717040782661861080197, 6.68153556271997047573220350467, 7.64886083962978640070870836781, 9.331685161582105759232125353483, 10.10813730839214742739571656591, 11.28529017363560569519632870070, 12.73417000688227013606079689177, 13.35192907216114017312694474724
