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
−14.85523291307456483136910499322, −13.70658305384352902670503706588, −12.18408298768799662981066093357, −11.60108896839824569354930802245, −10.11439729708169181782943475187, −9.043075354512368013186100321798, −7.67783469617989167820368840851, −5.84665860595227451021323068033, −4.84334816865138476903176497857, −2.53615521791389494222022443180,
0.789650534567216147754534854108, 3.54571937885128654035376213578, 5.33915138605416318804930165410, 7.01661750108004624773387249594, 7.77605532222556755936756068046, 9.796711792669717987335971733992, 10.57266548753053579336337603484, 12.05026963315747231437508489001, 12.86580561560083338888035775079, 14.11979687926607388830021377210