L(s) = 1 | + (−0.587 + 1.28i)2-s + (−2.19 − 0.644i)3-s + (−1.30 − 1.51i)4-s + (−1.20 − 1.88i)5-s + (2.11 − 2.44i)6-s + (−4.60 − 0.661i)7-s + (2.71 − 0.796i)8-s + (1.88 + 1.21i)9-s + (3.13 − 0.450i)10-s + (1.90 + 4.16i)12-s + (3.55 − 5.53i)14-s + (1.44 + 4.91i)15-s + (−0.569 + 3.95i)16-s + (−2.66 + 1.71i)18-s + (−1.25 + 4.29i)20-s + (9.68 + 4.42i)21-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)2-s + (−1.26 − 0.372i)3-s + (−0.654 − 0.755i)4-s + (−0.540 − 0.841i)5-s + (0.865 − 0.998i)6-s + (−1.73 − 0.250i)7-s + (0.959 − 0.281i)8-s + (0.628 + 0.403i)9-s + (0.989 − 0.142i)10-s + (0.548 + 1.20i)12-s + (0.950 − 1.47i)14-s + (0.372 + 1.26i)15-s + (−0.142 + 0.989i)16-s + (−0.628 + 0.403i)18-s + (−0.281 + 0.959i)20-s + (2.11 + 0.964i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.142284 + 0.161174i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.142284 + 0.161174i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 - 1.28i)T \) |
| 5 | \( 1 + (1.20 + 1.88i)T \) |
| 23 | \( 1 + (-4.02 - 2.61i)T \) |
good | 3 | \( 1 + (2.19 + 0.644i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (4.60 + 0.661i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.138 + 0.160i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (2.28 - 1.46i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.29 + 11.2i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-4.17 - 14.2i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-14.8 - 6.79i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-7.29 + 11.3i)T + (-34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (3.63 - 12.3i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (40.2 - 88.2i)T^{2} \) |
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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
−11.28998876412376914471719420237, −10.25274999658284305759513861965, −9.444533974191345940562086216885, −8.583886173903516216813363761057, −7.29574672002322375556375054657, −6.70570977244104195376380424264, −5.81111785464979688296106253275, −5.02282222414513775932461417352, −3.70757439091975939133553046215, −0.867060846572730230991179672088,
0.25911758290594517926183951773, 2.78854606030758309391560103120, 3.65581613939602082832441242237, 4.90156448748716538319958730048, 6.26267232529149480244627417197, 6.90011318966078935884145727470, 8.240653701517378459411390781704, 9.554419347717378730005571872717, 10.02732863738980554047953931219, 10.97475343173541507928426481093