L(s) = 1 | + (−2.59 − 1.5i)3-s + (2.59 − 1.5i)7-s + (3 + 5.19i)9-s + (−1.5 + 2.59i)11-s + (3.46 − i)13-s + (−6.06 + 3.5i)17-s + (0.5 + 0.866i)19-s − 9·21-s + (−6.06 − 3.5i)23-s − 9i·27-s + (−2.5 + 4.33i)29-s − 4·31-s + (7.79 − 4.5i)33-s + (2.59 + 1.5i)37-s + (−10.5 − 2.59i)39-s + ⋯ |
L(s) = 1 | + (−1.49 − 0.866i)3-s + (0.981 − 0.566i)7-s + (1 + 1.73i)9-s + (−0.452 + 0.783i)11-s + (0.960 − 0.277i)13-s + (−1.47 + 0.848i)17-s + (0.114 + 0.198i)19-s − 1.96·21-s + (−1.26 − 0.729i)23-s − 1.73i·27-s + (−0.464 + 0.804i)29-s − 0.718·31-s + (1.35 − 0.783i)33-s + (0.427 + 0.246i)37-s + (−1.68 − 0.416i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5942707869\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5942707869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.46 + i)T \) |
good | 3 | \( 1 + (2.59 + 1.5i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (6.06 - 3.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.79 + 4.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.2 + 6.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.52 - 5.5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.24186344233696307244246316081, −8.808088525392697477932251842545, −7.88491431042209290981448554964, −7.31585496887357908714642008563, −6.38126906711376896659482517145, −5.80498441206477212824978277536, −4.77267191781205223333612541309, −4.13405823844536446109174462772, −2.09626907368375395436704493030, −1.23707753508114821971067660253,
0.32729642532591563108742734981, 2.03174172627228209767222686356, 3.72000397069170886026498023035, 4.53339359385173684778176438070, 5.40171219128121160443062457696, 5.86826651645782380958003358713, 6.76546306165468248768041626248, 7.957230359615789754279522098452, 8.868527764200207336072696153027, 9.561433960866115018130088914855