L(s) = 1 | + (1 + i)2-s + (−1.36 + 1.36i)3-s + 2i·4-s + (−3.17 − 3.86i)5-s − 2.72·6-s + (7.21 + 7.21i)7-s + (−2 + 2i)8-s + 5.29i·9-s + (0.691 − 7.03i)10-s − 14.8·11-s + (−2.72 − 2.72i)12-s + (−4.91 + 4.91i)13-s + 14.4i·14-s + (9.57 + 0.941i)15-s − 4·16-s + (−17.8 − 17.8i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (−0.453 + 0.453i)3-s + 0.5i·4-s + (−0.634 − 0.772i)5-s − 0.453·6-s + (1.03 + 1.03i)7-s + (−0.250 + 0.250i)8-s + 0.588i·9-s + (0.0691 − 0.703i)10-s − 1.34·11-s + (−0.226 − 0.226i)12-s + (−0.377 + 0.377i)13-s + 1.03i·14-s + (0.638 + 0.0627i)15-s − 0.250·16-s + (−1.04 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.181275 + 1.08924i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.181275 + 1.08924i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 + (3.17 + 3.86i)T \) |
| 23 | \( 1 + (3.39 - 3.39i)T \) |
good | 3 | \( 1 + (1.36 - 1.36i)T - 9iT^{2} \) |
| 7 | \( 1 + (-7.21 - 7.21i)T + 49iT^{2} \) |
| 11 | \( 1 + 14.8T + 121T^{2} \) |
| 13 | \( 1 + (4.91 - 4.91i)T - 169iT^{2} \) |
| 17 | \( 1 + (17.8 + 17.8i)T + 289iT^{2} \) |
| 19 | \( 1 - 29.6iT - 361T^{2} \) |
| 29 | \( 1 + 11.0iT - 841T^{2} \) |
| 31 | \( 1 - 42.3T + 961T^{2} \) |
| 37 | \( 1 + (-14.7 - 14.7i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 23.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-0.419 + 0.419i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-53.0 - 53.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (58.3 - 58.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 13.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 39.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + (16.9 + 16.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 103.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-41.5 + 41.5i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 155. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-12.7 + 12.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 141. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (29.4 + 29.4i)T + 9.40e3iT^{2} \) |
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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
−12.24700116513579883550273364562, −11.66544766923847093109492523133, −10.74408780500122409375658643815, −9.338387415303841278853875731044, −8.120114627370094902119858347466, −7.75138950313583032301988395001, −5.86863426322771119080410872347, −4.97667931117871095104984899602, −4.48500638316649469321961188967, −2.39554648866499812083991091566,
0.51972224347904153347236124230, 2.49708523012231786198323589377, 3.99798075929189135814454529576, 5.03230963565957046736233204548, 6.51901985802673595845716900290, 7.35056963224485049386170024898, 8.361173341113829446140879992087, 10.11461792433378073739363093932, 11.00101895203364225937239134179, 11.28331719494161076959792003112