L(s) = 1 | + (1.67 + 1.67i)2-s + (−1.56 − 2.55i)3-s + 1.58i·4-s + (0.529 − 4.97i)5-s + (1.66 − 6.89i)6-s + (6.78 + 1.73i)7-s + (4.03 − 4.03i)8-s + (−4.10 + 8.01i)9-s + (9.19 − 7.42i)10-s − 4.41i·11-s + (4.06 − 2.48i)12-s + (1.62 − 1.62i)13-s + (8.43 + 14.2i)14-s + (−13.5 + 6.42i)15-s + 19.8·16-s + (−13.9 + 13.9i)17-s + ⋯ |
L(s) = 1 | + (0.835 + 0.835i)2-s + (−0.521 − 0.853i)3-s + 0.397i·4-s + (0.105 − 0.994i)5-s + (0.277 − 1.14i)6-s + (0.968 + 0.247i)7-s + (0.503 − 0.503i)8-s + (−0.455 + 0.890i)9-s + (0.919 − 0.742i)10-s − 0.401i·11-s + (0.338 − 0.207i)12-s + (0.124 − 0.124i)13-s + (0.602 + 1.01i)14-s + (−0.903 + 0.428i)15-s + 1.23·16-s + (−0.819 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.84532 - 0.337128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84532 - 0.337128i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.56 + 2.55i)T \) |
| 5 | \( 1 + (-0.529 + 4.97i)T \) |
| 7 | \( 1 + (-6.78 - 1.73i)T \) |
good | 2 | \( 1 + (-1.67 - 1.67i)T + 4iT^{2} \) |
| 11 | \( 1 + 4.41iT - 121T^{2} \) |
| 13 | \( 1 + (-1.62 + 1.62i)T - 169iT^{2} \) |
| 17 | \( 1 + (13.9 - 13.9i)T - 289iT^{2} \) |
| 19 | \( 1 + 0.694T + 361T^{2} \) |
| 23 | \( 1 + (23.1 - 23.1i)T - 529iT^{2} \) |
| 29 | \( 1 - 49.1T + 841T^{2} \) |
| 31 | \( 1 - 33.8iT - 961T^{2} \) |
| 37 | \( 1 + (-2.02 + 2.02i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 32.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (30.4 + 30.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (18.7 - 18.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (33.8 - 33.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 23.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (56.3 - 56.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 92.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-95.4 + 95.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 100. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-5.62 - 5.62i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 158. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (37.1 + 37.1i)T + 9.40e3iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.58386648383985942782945000834, −12.64762422805875175796743592507, −11.77321744406308502013401763805, −10.50507364578969081015012323618, −8.607559204015342872479168001424, −7.75891484916194769642514942701, −6.33958568192421526639674381710, −5.46017841559173339397842820466, −4.48937429597419750367048833137, −1.46937766676904354976203739990,
2.50464403759182809581020598010, 4.05647938009062051802581766029, 4.92450209435073703139195342403, 6.49874662273754251975539691515, 8.097273200089719597171510574648, 9.845048913521335069840096346511, 10.79745985460534712793587322568, 11.38965987226757882890989373912, 12.21375175214651482363152541033, 13.73576619803063591846869225789