L(s) = 1 | + (−2.56 − 2.56i)3-s + (3.91 − 55.7i)5-s + (86.5 − 86.5i)7-s − 229. i·9-s − 138. i·11-s + (−393. + 393. i)13-s + (−153. + 133. i)15-s + (724. + 724. i)17-s + 1.46e3·19-s − 444.·21-s + (−2.85e3 − 2.85e3i)23-s + (−3.09e3 − 436. i)25-s + (−1.21e3 + 1.21e3i)27-s + 2.51e3i·29-s − 2.96e3i·31-s + ⋯ |
L(s) = 1 | + (−0.164 − 0.164i)3-s + (0.0700 − 0.997i)5-s + (0.667 − 0.667i)7-s − 0.945i·9-s − 0.344i·11-s + (−0.645 + 0.645i)13-s + (−0.175 + 0.152i)15-s + (0.607 + 0.607i)17-s + 0.930·19-s − 0.220·21-s + (−1.12 − 1.12i)23-s + (−0.990 − 0.139i)25-s + (−0.320 + 0.320i)27-s + 0.555i·29-s − 0.554i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.370318258\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.370318258\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-3.91 + 55.7i)T \) |
good | 3 | \( 1 + (2.56 + 2.56i)T + 243iT^{2} \) |
| 7 | \( 1 + (-86.5 + 86.5i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 138. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (393. - 393. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-724. - 724. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 1.46e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.85e3 + 2.85e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 2.51e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.96e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (1.45e3 + 1.45e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.78e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (4.66e3 + 4.66e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-2.07e4 + 2.07e4i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.15e4 - 1.15e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 3.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (1.53e4 - 1.53e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 7.24e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (6.69e3 - 6.69e3i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 3.66e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.57e4 - 6.57e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 5.58e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-5.01e4 - 5.01e4i)T + 8.58e9iT^{2} \) |
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
−11.95046739437152782283517462993, −10.51560671364558944580165107127, −9.464998810279881079635293819151, −8.456966961927042961726753773677, −7.41065215336081508619196474858, −6.09442957517956192276926409519, −4.87075567730748632818579412078, −3.75655139210449209207278569361, −1.63460240928748611426302421262, −0.45682829701277687483776477033,
1.92116645520560213442555609935, 3.13437255440067169987592080004, 4.92067533599916355569011486231, 5.78325632208015428824836835390, 7.37411177394028334206244005608, 7.991801532383061857402899116966, 9.637013350230052253851945740713, 10.36361617430524381740779728318, 11.46489722289101973694772766571, 12.09923309053950767034113082857