L(s) = 1 | + (−1.98 − 1.98i)3-s + (−0.596 + 0.596i)5-s − 29.0i·7-s − 19.1i·9-s + (−12.1 + 12.1i)11-s + (−48.5 − 48.5i)13-s + 2.36·15-s + 86.7·17-s + (54.8 + 54.8i)19-s + (−57.6 + 57.6i)21-s − 70.2i·23-s + 124. i·25-s + (−91.5 + 91.5i)27-s + (63.4 + 63.4i)29-s + 8.86·31-s + ⋯ |
L(s) = 1 | + (−0.381 − 0.381i)3-s + (−0.0533 + 0.0533i)5-s − 1.57i·7-s − 0.708i·9-s + (−0.332 + 0.332i)11-s + (−1.03 − 1.03i)13-s + 0.0407·15-s + 1.23·17-s + (0.662 + 0.662i)19-s + (−0.599 + 0.599i)21-s − 0.636i·23-s + 0.994i·25-s + (−0.652 + 0.652i)27-s + (0.405 + 0.405i)29-s + 0.0513·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.671504 - 0.828081i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.671504 - 0.828081i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (1.98 + 1.98i)T + 27iT^{2} \) |
| 5 | \( 1 + (0.596 - 0.596i)T - 125iT^{2} \) |
| 7 | \( 1 + 29.0iT - 343T^{2} \) |
| 11 | \( 1 + (12.1 - 12.1i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (48.5 + 48.5i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 86.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-54.8 - 54.8i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 70.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-63.4 - 63.4i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 8.86T + 2.97e4T^{2} \) |
| 37 | \( 1 + (21.7 - 21.7i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 153. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-120. + 120. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 99.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-389. + 389. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-324. + 324. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (0.339 + 0.339i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (565. + 565. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 419. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 374. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 705.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-947. - 947. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 4.72iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 379.T + 9.12e5T^{2} \) |
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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
−14.11979687926607388830021377210, −12.86580561560083338888035775079, −12.05026963315747231437508489001, −10.57266548753053579336337603484, −9.796711792669717987335971733992, −7.77605532222556755936756068046, −7.01661750108004624773387249594, −5.33915138605416318804930165410, −3.54571937885128654035376213578, −0.789650534567216147754534854108,
2.53615521791389494222022443180, 4.84334816865138476903176497857, 5.84665860595227451021323068033, 7.67783469617989167820368840851, 9.043075354512368013186100321798, 10.11439729708169181782943475187, 11.60108896839824569354930802245, 12.18408298768799662981066093357, 13.70658305384352902670503706588, 14.85523291307456483136910499322