L(s) = 1 | + (2 − 2i)2-s + 15.0i·3-s − 8i·4-s + (30.6 + 30.6i)5-s + (30.1 + 30.1i)6-s − 89.3·7-s + (−16 − 16i)8-s − 146.·9-s + 122.·10-s − 96.8i·11-s + 120.·12-s + (78.2 + 78.2i)13-s + (−178. + 178. i)14-s + (−462. + 462. i)15-s − 64·16-s + (79.3 + 79.3i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + 1.67i·3-s − 0.5i·4-s + (1.22 + 1.22i)5-s + (0.838 + 0.838i)6-s − 1.82·7-s + (−0.250 − 0.250i)8-s − 1.81·9-s + 1.22·10-s − 0.800i·11-s + 0.838·12-s + (0.463 + 0.463i)13-s + (−0.911 + 0.911i)14-s + (−2.05 + 2.05i)15-s − 0.250·16-s + (0.274 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.17996 + 1.58425i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17996 + 1.58425i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 + 2i)T \) |
| 37 | \( 1 + (-498. - 1.27e3i)T \) |
good | 3 | \( 1 - 15.0iT - 81T^{2} \) |
| 5 | \( 1 + (-30.6 - 30.6i)T + 625iT^{2} \) |
| 7 | \( 1 + 89.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 96.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-78.2 - 78.2i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-79.3 - 79.3i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + (-438. - 438. i)T + 1.30e5iT^{2} \) |
| 23 | \( 1 + (-52.6 - 52.6i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-631. + 631. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-225. + 225. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + 39.5iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (1.19e3 + 1.19e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 411.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 3.84e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (854. + 854. i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (730. - 730. i)T - 1.38e7iT^{2} \) |
| 67 | \( 1 + 634. iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.01e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 5.16e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (6.52e3 + 6.52e3i)T + 3.89e7iT^{2} \) |
| 83 | \( 1 - 3.91e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-1.62e3 + 1.62e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (-6.80e3 - 6.80e3i)T + 8.85e7iT^{2} \) |
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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
−14.04809103901983148431474504564, −13.45559402431141241005043279231, −11.68059716454085295824232837775, −10.33564337038104431858990534341, −10.07631747457069410193263374774, −9.219631445144592853710019838315, −6.36834144460863802559233735694, −5.69250864548255536131276540583, −3.62583029797854442425644190205, −2.97821727296221870851908581571,
0.899151816340347699194058690776, 2.72581795509727457304634876415, 5.34085663620542474097425996958, 6.34509640416012039096334060275, 7.20165390453287218438526237352, 8.778773455022981469688470211921, 9.758529756302020684006813156445, 12.11317492940465314870448409798, 12.86039296220378387806064984260, 13.23295958712334331718156796997