| L(s) = 1 | + (−0.613 − 0.800i)2-s + (0.0530 − 1.73i)3-s + (0.254 − 0.949i)4-s + (3.54 + 0.232i)5-s + (−1.41 + 1.02i)6-s + (0.00291 + 0.0445i)7-s + (−2.77 + 1.15i)8-s + (−2.99 − 0.183i)9-s + (−1.99 − 2.98i)10-s + (4.54 + 3.98i)11-s + (−1.63 − 0.490i)12-s + (−3.51 − 0.940i)13-s + (0.0338 − 0.0296i)14-s + (0.590 − 6.12i)15-s + (0.924 + 0.533i)16-s + (−3.77 − 1.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.434 − 0.565i)2-s + (0.0306 − 0.999i)3-s + (0.127 − 0.474i)4-s + (1.58 + 0.103i)5-s + (−0.578 + 0.416i)6-s + (0.00110 + 0.0168i)7-s + (−0.982 + 0.407i)8-s + (−0.998 − 0.0611i)9-s + (−0.629 − 0.942i)10-s + (1.37 + 1.20i)11-s + (−0.470 − 0.141i)12-s + (−0.973 − 0.260i)13-s + (0.00904 − 0.00793i)14-s + (0.152 − 1.58i)15-s + (0.231 + 0.133i)16-s + (−0.914 − 0.404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.683651 - 0.880901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.683651 - 0.880901i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0530 + 1.73i)T \) |
| 17 | \( 1 + (3.77 + 1.66i)T \) |
| good | 2 | \( 1 + (0.613 + 0.800i)T + (-0.517 + 1.93i)T^{2} \) |
| 5 | \( 1 + (-3.54 - 0.232i)T + (4.95 + 0.652i)T^{2} \) |
| 7 | \( 1 + (-0.00291 - 0.0445i)T + (-6.94 + 0.913i)T^{2} \) |
| 11 | \( 1 + (-4.54 - 3.98i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (3.51 + 0.940i)T + (11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (2.46 + 1.02i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.31 - 6.82i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (-0.0353 + 0.0174i)T + (17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-2.74 - 3.13i)T + (-4.04 + 30.7i)T^{2} \) |
| 37 | \( 1 + (1.01 + 5.09i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-2.12 + 4.30i)T + (-24.9 - 32.5i)T^{2} \) |
| 43 | \( 1 + (0.107 - 0.814i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (1.62 - 0.435i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.17 + 2.83i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (7.99 + 6.13i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (0.0920 - 0.00603i)T + (60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (5.04 - 2.91i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.65 + 1.12i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (6.75 + 4.51i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-3.54 + 4.04i)T + (-10.3 - 78.3i)T^{2} \) |
| 83 | \( 1 + (-4.62 + 3.54i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (12.7 - 12.7i)T - 89iT^{2} \) |
| 97 | \( 1 + (-1.60 - 3.25i)T + (-59.0 + 76.9i)T^{2} \) |
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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
−12.58034568018885197249511777215, −11.73900368410127383101171077308, −10.59206154914803601502633230836, −9.463986040007513046725506904027, −9.105062034929366911127220153102, −7.11144029195743469551985826563, −6.36272289576428696817889492380, −5.21947504781395389421292896110, −2.44175657441826020968316640226, −1.60459644646666084148551493204,
2.69792758899439081003852382452, 4.35537033768961610831862606371, 5.97593138041135256178041362039, 6.58779682040571446604166793425, 8.544338232231612902363741100744, 9.051100242412368592962927403857, 9.929137731520566990538620402520, 11.05168028307650469359426364120, 12.22711010116110266009898385237, 13.49727740187316213044237016460
