L(s) = 1 | + (2 − 3.46i)2-s + (15.2 + 3.37i)3-s + (−7.99 − 13.8i)4-s + (−52.1 − 90.3i)5-s + (42.1 − 45.9i)6-s + (64.3 − 111. i)7-s − 63.9·8-s + (220. + 102. i)9-s − 417.·10-s + (−60.5 + 104. i)11-s + (−74.9 − 237. i)12-s + (−230. − 399. i)13-s + (−257. − 445. i)14-s + (−488. − 1.55e3i)15-s + (−128 + 221. i)16-s − 63.6·17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.976 + 0.216i)3-s + (−0.249 − 0.433i)4-s + (−0.933 − 1.61i)5-s + (0.477 − 0.521i)6-s + (0.496 − 0.859i)7-s − 0.353·8-s + (0.906 + 0.423i)9-s − 1.32·10-s + (−0.150 + 0.261i)11-s + (−0.150 − 0.476i)12-s + (−0.378 − 0.656i)13-s + (−0.350 − 0.607i)14-s + (−0.561 − 1.78i)15-s + (−0.125 + 0.216i)16-s − 0.0534·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0876i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.160004264\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160004264\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 + (-15.2 - 3.37i)T \) |
| 11 | \( 1 + (60.5 - 104. i)T \) |
good | 5 | \( 1 + (52.1 + 90.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-64.3 + 111. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 13 | \( 1 + (230. + 399. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 63.6T + 1.41e6T^{2} \) |
| 19 | \( 1 + 110.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (321. + 556. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.85e3 - 3.21e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (786. + 1.36e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 6.91e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (8.72e3 + 1.51e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (7.32e3 - 1.26e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.03e4 + 1.78e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 2.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.60e3 - 2.78e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.58e4 + 4.47e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.40e4 + 4.16e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.43e4 - 2.49e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.59e3 - 6.22e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 5.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.30e4 + 3.98e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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
−11.14899085213860547044179075556, −10.07414396226450200480883709329, −9.054114164173774709046950795202, −8.170511892425522741092815661440, −7.42851896989875760620920087832, −5.10481059248837013414050795475, −4.41093352910252236591553037650, −3.49169986945964811544503792260, −1.72250731360519324854433647056, −0.50883315275035854429836158132,
2.30870289131199874406493691051, 3.26251281215237710850271113233, 4.38567543196213644264176489969, 6.15310820619497696547157051416, 7.16278521965404806844990343279, 7.85278369876222080102577970010, 8.768480960040047188593288301569, 10.03520332301531493143510981354, 11.37842984344395461880886533441, 12.02539426032974786466029747428