L(s) = 1 | + (53.0 − 17.7i)5-s − 188. i·7-s − 501.·11-s + 1.06e3i·13-s − 29.5i·17-s + 1.57e3·19-s − 1.29e3i·23-s + (2.49e3 − 1.88e3i)25-s − 3.58e3·29-s + 3.52e3·31-s + (−3.35e3 − 1.00e4i)35-s − 8.41e3i·37-s − 7.01e3·41-s − 2.26e4i·43-s + 3.50e3i·47-s + ⋯ |
L(s) = 1 | + (0.948 − 0.317i)5-s − 1.45i·7-s − 1.25·11-s + 1.74i·13-s − 0.0248i·17-s + 1.00·19-s − 0.510i·23-s + (0.798 − 0.602i)25-s − 0.791·29-s + 0.659·31-s + (−0.463 − 1.38i)35-s − 1.01i·37-s − 0.651·41-s − 1.87i·43-s + 0.231i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.138196341\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.138196341\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-53.0 + 17.7i)T \) |
good | 7 | \( 1 + 188. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 501.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.06e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 29.5iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.41e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.26e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.50e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.73e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.02e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.76e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.99e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.84e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.10e5iT - 8.58e9T^{2} \) |
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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
−9.421760270235857254935757879211, −8.457688763151813699194836016865, −7.33285876419298077946466711820, −6.79354132812600344602976107006, −5.59774254518515264310342046061, −4.73063168591573360327359633312, −3.80027660480860817256968660284, −2.39633392741632630567932788603, −1.39368699581079913726855152068, −0.22106664779076473625456166777,
1.38416421474668904496555765184, 2.73804949833829672232757428643, 2.98641798392962560008580654481, 5.10624140582134474995688137501, 5.50516293887539343982221290495, 6.23374320655105945572158237837, 7.61089391307246936319448055704, 8.259686572781381981065565044938, 9.319666731975254480169046148848, 9.979943380394942567915323417220