L(s) = 1 | + (−1.76 − 3.05i)2-s + (−2.20 + 3.82i)4-s + (2.5 − 4.33i)5-s + (−12.7 − 22.0i)7-s − 12.6·8-s − 17.6·10-s + (−35.6 − 61.7i)11-s + (25.6 − 44.5i)13-s + (−44.8 + 77.6i)14-s + (39.9 + 69.1i)16-s + 33.3·17-s + 113.·19-s + (11.0 + 19.1i)20-s + (−125. + 217. i)22-s + (40.9 − 70.9i)23-s + ⋯ |
L(s) = 1 | + (−0.622 − 1.07i)2-s + (−0.275 + 0.477i)4-s + (0.223 − 0.387i)5-s + (−0.686 − 1.18i)7-s − 0.558·8-s − 0.557·10-s + (−0.977 − 1.69i)11-s + (0.548 − 0.949i)13-s + (−0.855 + 1.48i)14-s + (0.623 + 1.08i)16-s + 0.475·17-s + 1.36·19-s + (0.123 + 0.213i)20-s + (−1.21 + 2.11i)22-s + (0.371 − 0.643i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8308720456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8308720456\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
good | 2 | \( 1 + (1.76 + 3.05i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (12.7 + 22.0i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (35.6 + 61.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-25.6 + 44.5i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-40.9 + 70.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (123. + 213. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (111. - 192. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 22.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + (217. - 376. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-118. - 205. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (53.9 + 93.5i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-85.5 + 148. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-39.7 - 68.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-305. + 529. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 511.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 410.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-396. - 687. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-135. - 233. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 177.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-440. - 763. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.28385937991410129459807178918, −9.584164044826230054534118330717, −8.499304509936457522686404844913, −7.72441303760432618218318601144, −6.23421076406775137938820296332, −5.36410453742457851338419371288, −3.53019946053603110106015213336, −2.97379100117213376636589282787, −1.06687844703190379614673497987, −0.39970941342470418892747316575,
2.09318911140121576853238322314, 3.34114668211825418911553238893, 5.24408411251321696149708634830, 5.89094596419186055486279964892, 7.15072220435847594494328370300, 7.42531089177034694833476130616, 8.868071607138967638027212132415, 9.392167631067759778698778488874, 10.14963379102608024839851223746, 11.58425882298017235719901093194