| L(s) = 1 | + (−1.16 + 0.801i)2-s + (1.71 − 2.23i)3-s + (0.715 − 1.86i)4-s + (1.90 − 1.46i)5-s + (−0.206 + 3.97i)6-s + (−0.262 + 2.63i)7-s + (0.663 + 2.74i)8-s + (−1.27 − 4.75i)9-s + (−1.04 + 3.22i)10-s + (2.49 − 0.328i)11-s + (−2.94 − 4.79i)12-s + (−1.76 + 0.729i)13-s + (−1.80 − 3.27i)14-s − 6.75i·15-s + (−2.97 − 2.67i)16-s + (−5.05 + 2.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.823 + 0.566i)2-s + (0.989 − 1.28i)3-s + (0.357 − 0.933i)4-s + (0.851 − 0.653i)5-s + (−0.0843 + 1.62i)6-s + (−0.0992 + 0.995i)7-s + (0.234 + 0.972i)8-s + (−0.424 − 1.58i)9-s + (−0.331 + 1.02i)10-s + (0.752 − 0.0991i)11-s + (−0.850 − 1.38i)12-s + (−0.488 + 0.202i)13-s + (−0.482 − 0.876i)14-s − 1.74i·15-s + (−0.744 − 0.667i)16-s + (−1.22 + 0.708i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16707 - 0.462076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16707 - 0.462076i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 - 0.801i)T \) |
| 7 | \( 1 + (0.262 - 2.63i)T \) |
| good | 3 | \( 1 + (-1.71 + 2.23i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (-1.90 + 1.46i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-2.49 + 0.328i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (1.76 - 0.729i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (5.05 - 2.92i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.908 + 6.90i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-0.497 - 1.85i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.54 - 8.55i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (2.59 + 4.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.70 - 2.07i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (4.86 - 4.86i)T - 41iT^{2} \) |
| 43 | \( 1 + (-1.94 + 4.70i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.40 - 1.96i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.89 + 0.644i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (0.694 + 5.27i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (7.60 + 1.00i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (4.18 - 5.45i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-7.62 - 7.62i)T + 71iT^{2} \) |
| 73 | \( 1 + (-10.2 - 2.73i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.49 - 4.32i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.84 - 4.07i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (3.81 - 1.02i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−12.34526210340253494458274367250, −11.20528043422321930971155232646, −9.508100292038419498620551447550, −8.955114411405752102472654712431, −8.466789197347320933613615024796, −7.07255174504882766314524454603, −6.43673289718916218890723688406, −5.18635513554530159232541928924, −2.50789328544009567299573775335, −1.53307775075580101045738112860,
2.24481411485934292941714729346, 3.46227742435361505838326144973, 4.42325136907701433035960985281, 6.52366434351821574596603471012, 7.67974457751982241142047254759, 8.840431402588327735415959437624, 9.662792704269017672340225263495, 10.23736211932823506521871539508, 10.80117176650505232680712966661, 12.13188059991355021365400352701
