L(s) = 1 | + 0.505·2-s + (0.985 − 2.83i)3-s − 3.74·4-s + (0.221 − 4.99i)5-s + (0.498 − 1.43i)6-s + 2.64i·7-s − 3.91·8-s + (−7.05 − 5.58i)9-s + (0.111 − 2.52i)10-s − 13.3i·11-s + (−3.68 + 10.6i)12-s + 3.27i·13-s + 1.33i·14-s + (−13.9 − 5.54i)15-s + 12.9·16-s + 21.9·17-s + ⋯ |
L(s) = 1 | + 0.252·2-s + (0.328 − 0.944i)3-s − 0.936·4-s + (0.0442 − 0.999i)5-s + (0.0830 − 0.238i)6-s + 0.377i·7-s − 0.489·8-s + (−0.784 − 0.620i)9-s + (0.0111 − 0.252i)10-s − 1.21i·11-s + (−0.307 + 0.884i)12-s + 0.251i·13-s + 0.0955i·14-s + (−0.929 − 0.369i)15-s + 0.812·16-s + 1.29·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.705398 - 1.04013i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.705398 - 1.04013i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.985 + 2.83i)T \) |
| 5 | \( 1 + (-0.221 + 4.99i)T \) |
| 7 | \( 1 - 2.64iT \) |
good | 2 | \( 1 - 0.505T + 4T^{2} \) |
| 11 | \( 1 + 13.3iT - 121T^{2} \) |
| 13 | \( 1 - 3.27iT - 169T^{2} \) |
| 17 | \( 1 - 21.9T + 289T^{2} \) |
| 19 | \( 1 + 4.36T + 361T^{2} \) |
| 23 | \( 1 - 30.4T + 529T^{2} \) |
| 29 | \( 1 - 4.90iT - 841T^{2} \) |
| 31 | \( 1 - 29.2T + 961T^{2} \) |
| 37 | \( 1 + 24.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 75.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 67.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 61.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 96.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 108. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 59.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 82.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 79.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 103. iT - 9.40e3T^{2} \) |
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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
−13.15791979597566689188450474101, −12.50988114379442623467831409588, −11.50102752781836767452962241374, −9.570728416055349207063206997736, −8.675213489691554085813651968627, −7.995241734518224844766220560440, −6.12639466709629084584951764705, −5.08166668281707612027765577423, −3.31176156404791037441289992106, −0.933629462429868711513831667804,
3.07442864920703614099260562907, 4.26656583127422417129111930703, 5.46887485951220361674062808806, 7.23653235530659263858000014642, 8.540027668160810134678726537657, 9.941698802442665486838678161277, 10.19769917726090616780923250375, 11.68931166609677123595633239499, 13.05248674265726213366304692464, 14.09240198899198240113086693708