L(s) = 1 | + 4.84i·3-s − 6.85·7-s − 14.4·9-s − 9.59·11-s − 10.0i·13-s − 27.8·17-s + (17.5 − 7.24i)19-s − 33.2i·21-s + 8.14·23-s − 26.3i·27-s − 17.8i·29-s − 20.3i·31-s − 46.4i·33-s + 65.1i·37-s + 48.8·39-s + ⋯ |
L(s) = 1 | + 1.61i·3-s − 0.979·7-s − 1.60·9-s − 0.872·11-s − 0.775i·13-s − 1.63·17-s + (0.924 − 0.381i)19-s − 1.58i·21-s + 0.354·23-s − 0.977i·27-s − 0.614i·29-s − 0.657i·31-s − 1.40i·33-s + 1.75i·37-s + 1.25·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9935837118\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9935837118\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-17.5 + 7.24i)T \) |
good | 3 | \( 1 - 4.84iT - 9T^{2} \) |
| 7 | \( 1 + 6.85T + 49T^{2} \) |
| 11 | \( 1 + 9.59T + 121T^{2} \) |
| 13 | \( 1 + 10.0iT - 169T^{2} \) |
| 17 | \( 1 + 27.8T + 289T^{2} \) |
| 23 | \( 1 - 8.14T + 529T^{2} \) |
| 29 | \( 1 + 17.8iT - 841T^{2} \) |
| 31 | \( 1 + 20.3iT - 961T^{2} \) |
| 37 | \( 1 - 65.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 9.60T + 2.20e3T^{2} \) |
| 53 | \( 1 - 14.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 0.442iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 65.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 103. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 88.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 86.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 47.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 58.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 165. iT - 9.40e3T^{2} \) |
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
−9.285192632048439652487629505853, −8.564655056930538238036203528221, −7.61672934397063406285527138508, −6.54560981400270001165718112084, −5.68748905543658667502015663588, −4.90442014654660277522982845669, −4.19744081861296947952005044438, −3.18429066234480584672604217356, −2.61423931102106358682533094570, −0.37068673450777400018090684105,
0.71306762003327888970109098828, 2.01261026526806198038880459075, 2.68828461316178018550371426406, 3.84029249936723395506791735058, 5.17361488154513290494411479021, 6.03318751689432983072369079673, 6.84863361635953144570058628887, 7.15866640754923858164640916900, 8.051792655708619736026296480690, 8.918707751369199713506247250739