| L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (3.49 + 1.02i)5-s + (0.415 − 0.909i)6-s + (−0.654 + 0.755i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (2.38 + 2.75i)10-s + (4.79 − 3.08i)11-s + (0.841 − 0.540i)12-s + (−1.87 − 2.16i)13-s + (−0.959 + 0.281i)14-s + (0.518 − 3.60i)15-s + (−0.654 + 0.755i)16-s + (1.55 − 3.39i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (1.56 + 0.458i)5-s + (0.169 − 0.371i)6-s + (−0.247 + 0.285i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.754 + 0.870i)10-s + (1.44 − 0.929i)11-s + (0.242 − 0.156i)12-s + (−0.520 − 0.600i)13-s + (−0.256 + 0.0752i)14-s + (0.133 − 0.930i)15-s + (−0.163 + 0.188i)16-s + (0.376 − 0.824i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.84309 + 0.419015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.84309 + 0.419015i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (3.25 - 3.51i)T \) |
| good | 5 | \( 1 + (-3.49 - 1.02i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-4.79 + 3.08i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.87 + 2.16i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.55 + 3.39i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.946 - 2.07i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.09 + 4.57i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.49 - 10.4i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (5.63 - 1.65i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (4.72 + 1.38i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.348 + 2.42i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + (0.195 - 0.225i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-7.80 - 9.00i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 12.9i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (9.78 + 6.28i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (7.55 + 4.85i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.39 - 3.05i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.93 - 10.3i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (3.26 - 0.960i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.40 - 9.78i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (14.7 + 4.32i)T + (81.6 + 52.4i)T^{2} \) |
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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
−9.979021500061981362771344104107, −9.237959792870331645310668085134, −8.366448999011931699822371160254, −7.13845651387119490838745590296, −6.54780678208499830485191532417, −5.76058625188295071409485735928, −5.28646360925378859828441762242, −3.57696778034941121863243358384, −2.68300200957855919850184209821, −1.46637473889654536724214635398,
1.46629994959858020293758129806, 2.39460506343629946213545804487, 3.91521880036212423685825268943, 4.57507708605437221509261591242, 5.58300947126550946183935097011, 6.31871505954721511116519030024, 7.09545910630584206747466020692, 8.771954644976224264940657894803, 9.424082225183307921758154957772, 9.967068729564581844959896352188
