| L(s) = 1 | + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (−1.63 + 3.58i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (4.05 + 1.19i)11-s + (0.959 + 0.281i)12-s + (0.877 + 1.92i)13-s + (−0.561 + 3.90i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (2.41 + 1.55i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.0636 − 0.442i)5-s + (0.343 + 0.220i)6-s + (−0.619 + 1.35i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.131 − 0.287i)10-s + (1.22 + 0.358i)11-s + (0.276 + 0.0813i)12-s + (0.243 + 0.532i)13-s + (−0.150 + 1.04i)14-s + (0.169 − 0.195i)15-s + (0.103 − 0.227i)16-s + (0.586 + 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.38483 + 0.758732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.38483 + 0.758732i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (1.66 + 4.49i)T \) |
| good | 7 | \( 1 + (1.63 - 3.58i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-4.05 - 1.19i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.877 - 1.92i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.41 - 1.55i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (2.20 - 1.41i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-8.55 - 5.49i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-6.42 + 7.41i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.362 - 2.51i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.513 + 3.56i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (8.19 + 9.45i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 8.95T + 47T^{2} \) |
| 53 | \( 1 + (4.05 - 8.88i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (2.31 + 5.07i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.622 + 0.718i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.480 + 0.140i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (10.2 - 3.00i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.02 + 1.94i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.98 - 4.33i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.12 + 14.7i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-11.7 - 13.5i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.06 + 14.3i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.45555220907690792351819979734, −9.681142691782240114120922733028, −8.874952465811448429780528845575, −8.253492964202003270311297999004, −6.62790330300925437748403287865, −6.09515355974173057832026699643, −4.91259208399235853930777682448, −4.05538467994113380643238609154, −3.00461023908571726893099654420, −1.80593356022470685628053271843,
1.16744505633458156076117598708, 3.04008715841965732634817453457, 3.64536916344806202048848186567, 4.72573649266309435772116244285, 6.41457560060528040627572669763, 6.53847292314587365970325855534, 7.61912324246835181820476078083, 8.374887529928479003455062800524, 9.702812222446825366942757098434, 10.34649139919518689761732558464
