L(s) = 1 | + (−1.5 − 2.59i)3-s + 2·5-s + (−0.5 + 0.866i)7-s + (−3 + 5.19i)9-s + (2.5 + 4.33i)11-s + (−1 − 3.46i)13-s + (−3 − 5.19i)15-s + (−1.5 + 2.59i)17-s + (1.5 − 2.59i)19-s + 3·21-s + (0.5 + 0.866i)23-s − 25-s + 9·27-s + (0.5 + 0.866i)29-s − 8·31-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.49i)3-s + 0.894·5-s + (−0.188 + 0.327i)7-s + (−1 + 1.73i)9-s + (0.753 + 1.30i)11-s + (−0.277 − 0.960i)13-s + (−0.774 − 1.34i)15-s + (−0.363 + 0.630i)17-s + (0.344 − 0.596i)19-s + 0.654·21-s + (0.104 + 0.180i)23-s − 0.200·25-s + 1.73·27-s + (0.0928 + 0.160i)29-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.642116 - 0.359827i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.642116 - 0.359827i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 3 | \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 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 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.5 + 9.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.21092882118191300765777380670, −13.84448869612692116459677524377, −12.78513831612051706202721323350, −12.24570811256434994037913864959, −10.83838465312126903038412118012, −9.367585541280490269000273252017, −7.55222493609191092871777397035, −6.48868611598529377211818851386, −5.38435142868249019827703093341, −1.93936325472051329137804869360,
3.82123330043846069705531138626, 5.35152479339514248803367851175, 6.46207850976247786168859201932, 9.043079211916414793675684041350, 9.803782251296111740342542992802, 10.94103692321319926642419367036, 11.81540459105864984067643132369, 13.68310616458939539746629441302, 14.56855894430709074688826922060, 16.03949682487103654508123102874