| L(s) = 1 | + (1.53 + 2.97i)3-s + (−0.875 + 0.0836i)5-s + (−0.805 + 2.52i)7-s + (−4.74 + 6.65i)9-s + (1.55 − 0.536i)11-s + (−0.786 − 2.67i)13-s + (−1.58 − 2.47i)15-s + (1.00 + 0.402i)17-s + (−2.68 + 1.07i)19-s + (−8.72 + 1.46i)21-s + (3.52 − 3.25i)23-s + (−4.15 + 0.799i)25-s + (−17.1 − 2.46i)27-s + (0.643 + 4.47i)29-s + (9.78 − 0.466i)31-s + ⋯ |
| L(s) = 1 | + (0.884 + 1.71i)3-s + (−0.391 + 0.0373i)5-s + (−0.304 + 0.952i)7-s + (−1.58 + 2.21i)9-s + (0.467 − 0.161i)11-s + (−0.218 − 0.742i)13-s + (−0.410 − 0.638i)15-s + (0.243 + 0.0976i)17-s + (−0.616 + 0.246i)19-s + (−1.90 + 0.319i)21-s + (0.735 − 0.677i)23-s + (−0.830 + 0.159i)25-s + (−3.29 − 0.473i)27-s + (0.119 + 0.831i)29-s + (1.75 − 0.0836i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.362069 + 1.54511i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.362069 + 1.54511i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.805 - 2.52i)T \) |
| 23 | \( 1 + (-3.52 + 3.25i)T \) |
| good | 3 | \( 1 + (-1.53 - 2.97i)T + (-1.74 + 2.44i)T^{2} \) |
| 5 | \( 1 + (0.875 - 0.0836i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-1.55 + 0.536i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (0.786 + 2.67i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-1.00 - 0.402i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (2.68 - 1.07i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.643 - 4.47i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-9.78 + 0.466i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-3.36 - 2.39i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-1.91 - 0.875i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.30 + 2.02i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (7.35 - 4.24i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.79 - 6.07i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (0.831 + 0.201i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-1.99 - 1.02i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (1.28 + 6.65i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (0.354 + 0.409i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (5.94 - 7.55i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-10.8 + 11.3i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-5.29 - 11.6i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.737 - 15.4i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (0.184 - 0.404i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.68700627657847575892835224822, −9.968032155005716049305100879908, −9.235932915132604774738992731871, −8.507030638160291930808586166087, −7.905002971096668099676064056905, −6.24885132800575379976666339030, −5.20510021083309239925599314755, −4.34281299763617008649667001969, −3.31990199210492783427010917474, −2.57326393746212454594069041554,
0.78134554872737276456644023871, 2.06150609426697157344399889959, 3.28369149385712289931584503574, 4.31749028956835044514817521519, 6.17670991650643536854769332534, 6.83301266563544896205206455522, 7.52134993774387030973920103833, 8.207832720217193086799151505284, 9.131950713165979999537465430869, 9.988920860789548454193777126484
