L(s) = 1 | + (1.20 − 2.90i)3-s + (3.98 − 1.65i)5-s + (22.4 − 22.4i)7-s + (12.1 + 12.1i)9-s + (16.5 + 39.8i)11-s + (−17.9 − 7.42i)13-s − 13.5i·15-s − 45.9i·17-s + (25.0 + 10.3i)19-s + (−38.0 − 91.9i)21-s + (−40.3 − 40.3i)23-s + (−75.2 + 75.2i)25-s + (128. − 53.0i)27-s + (88.6 − 214. i)29-s + 260.·31-s + ⋯ |
L(s) = 1 | + (0.231 − 0.558i)3-s + (0.356 − 0.147i)5-s + (1.20 − 1.20i)7-s + (0.448 + 0.448i)9-s + (0.452 + 1.09i)11-s + (−0.382 − 0.158i)13-s − 0.233i·15-s − 0.656i·17-s + (0.301 + 0.125i)19-s + (−0.395 − 0.955i)21-s + (−0.365 − 0.365i)23-s + (−0.601 + 0.601i)25-s + (0.912 − 0.378i)27-s + (0.567 − 1.37i)29-s + 1.50·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.473871815\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.473871815\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-1.20 + 2.90i)T + (-19.0 - 19.0i)T^{2} \) |
| 5 | \( 1 + (-3.98 + 1.65i)T + (88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (-22.4 + 22.4i)T - 343iT^{2} \) |
| 11 | \( 1 + (-16.5 - 39.8i)T + (-941. + 941. i)T^{2} \) |
| 13 | \( 1 + (17.9 + 7.42i)T + (1.55e3 + 1.55e3i)T^{2} \) |
| 17 | \( 1 + 45.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-25.0 - 10.3i)T + (4.85e3 + 4.85e3i)T^{2} \) |
| 23 | \( 1 + (40.3 + 40.3i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-88.6 + 214. i)T + (-1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 - 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (70.4 - 29.1i)T + (3.58e4 - 3.58e4i)T^{2} \) |
| 41 | \( 1 + (251. + 251. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (-95.7 - 231. i)T + (-5.62e4 + 5.62e4i)T^{2} \) |
| 47 | \( 1 - 15.5iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (171. + 414. i)T + (-1.05e5 + 1.05e5i)T^{2} \) |
| 59 | \( 1 + (53.3 - 22.0i)T + (1.45e5 - 1.45e5i)T^{2} \) |
| 61 | \( 1 + (297. - 718. i)T + (-1.60e5 - 1.60e5i)T^{2} \) |
| 67 | \( 1 + (-377. + 911. i)T + (-2.12e5 - 2.12e5i)T^{2} \) |
| 71 | \( 1 + (359. - 359. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-605. - 605. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 380. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (235. + 97.4i)T + (4.04e5 + 4.04e5i)T^{2} \) |
| 89 | \( 1 + (949. - 949. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 - 663.T + 9.12e5T^{2} \) |
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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
−11.53883568984400398307793618803, −10.32721791777941431775911192193, −9.726444113420134691396307673798, −8.163599951471774657403658419562, −7.54306157662707307296584766285, −6.71237753464922633879791834243, −4.98745408751426364065991922440, −4.24691856411211311049507229142, −2.18694483461073096583717441599, −1.13506793449928699263278909570,
1.54149296422978240291284213872, 3.03521859094500460035533130604, 4.44149312171512819826123237234, 5.53306241596003897907657869861, 6.55157633725949712908805911797, 8.152168095889913263063689538125, 8.794799437486847927988825833911, 9.731242053435353540890532391435, 10.75538118867875550903558521494, 11.75723030581709358142633358984