L(s) = 1 | + (−0.671 + 0.775i)3-s + (0.142 − 0.989i)5-s + (−1.80 − 3.94i)7-s + (0.277 + 1.92i)9-s + (−6.12 + 1.79i)11-s + (−0.327 + 0.716i)13-s + (0.671 + 0.775i)15-s + (−5.62 + 3.61i)17-s + (0.285 + 0.183i)19-s + (4.26 + 1.25i)21-s + (2.98 − 3.75i)23-s + (−0.959 − 0.281i)25-s + (−4.27 − 2.74i)27-s + (−3.76 + 2.41i)29-s + (−5.28 − 6.10i)31-s + ⋯ |
L(s) = 1 | + (−0.387 + 0.447i)3-s + (0.0636 − 0.442i)5-s + (−0.680 − 1.49i)7-s + (0.0923 + 0.642i)9-s + (−1.84 + 0.542i)11-s + (−0.0907 + 0.198i)13-s + (0.173 + 0.200i)15-s + (−1.36 + 0.876i)17-s + (0.0655 + 0.0421i)19-s + (0.931 + 0.273i)21-s + (0.622 − 0.782i)23-s + (−0.191 − 0.0563i)25-s + (−0.821 − 0.528i)27-s + (−0.698 + 0.449i)29-s + (−0.949 − 1.09i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0101089 - 0.0968188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0101089 - 0.0968188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (-2.98 + 3.75i)T \) |
good | 3 | \( 1 + (0.671 - 0.775i)T + (-0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (1.80 + 3.94i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (6.12 - 1.79i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.327 - 0.716i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (5.62 - 3.61i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.285 - 0.183i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (3.76 - 2.41i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (5.28 + 6.10i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.127 + 0.886i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.293 + 2.04i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.23 + 2.58i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + (3.77 + 8.27i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (5.63 - 12.3i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-4.70 - 5.43i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.54 - 0.453i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (5.04 + 1.48i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.83 - 1.17i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.756 + 1.65i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.323 + 2.24i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-3.41 + 3.93i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.45 + 10.1i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.58198902065073601648214563812, −10.08600596195460838944339660489, −8.930305119329731122478840160975, −7.71309439152229082961570155580, −7.10683204901418482863314935786, −5.75268204837835552843286589563, −4.71977773662925490053387186446, −3.99987664108022744667616731916, −2.27703947632006885296477426591, −0.05732086137058969371431932891,
2.39687590777480535246968203987, 3.21504614640072639248056107552, 5.17389173777379361805819566683, 5.83550095615147535269911333997, 6.76215734676148812191963481409, 7.71552409430583491406105223694, 8.949578570417263443818744365235, 9.516440763962085018820116520392, 10.80226606373494396886884827171, 11.41825747918814343640410129387