L(s) = 1 | − 23.0i·2-s − 78.5i·3-s − 273.·4-s − 548. i·5-s − 1.80e3·6-s − 2.21e3i·7-s + 399. i·8-s + 395.·9-s − 1.26e4·10-s − 2.23e4·11-s + 2.14e4i·12-s + 2.57e4·13-s − 5.09e4·14-s − 4.30e4·15-s − 6.07e4·16-s + 9.48e4·17-s + ⋯ |
L(s) = 1 | − 1.43i·2-s − 0.969i·3-s − 1.06·4-s − 0.877i·5-s − 1.39·6-s − 0.922i·7-s + 0.0974i·8-s + 0.0602·9-s − 1.26·10-s − 1.52·11-s + 1.03i·12-s + 0.900·13-s − 1.32·14-s − 0.850·15-s − 0.927·16-s + 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.08684 + 1.25695i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08684 + 1.25695i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (4.93e5 + 3.38e6i)T \) |
good | 2 | \( 1 + 23.0iT - 256T^{2} \) |
| 3 | \( 1 + 78.5iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 548. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.21e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.23e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 2.57e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 9.48e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.37e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.44e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 8.43e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 5.57e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.34e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.42e6T + 7.98e12T^{2} \) |
| 47 | \( 1 + 4.12e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 7.17e5T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.58e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + 1.76e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 2.33e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.27e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.05e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 6.41e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.42e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 8.42e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 6.90e7T + 7.83e15T^{2} \) |
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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
−13.02821198901518315284058254242, −12.40579891931471764266738756568, −10.99052225010123362616607354500, −10.05908650050203727055350907243, −8.428690528119905551447327597230, −7.07788056571071795404360880724, −4.91144287658711010729491143385, −3.18804168596234580275011452836, −1.43238974412245003418600575797, −0.69651122987050327627663698659,
2.96773036168082289632937702054, 4.95350877033162755873970097051, 5.98106276572318579088839323892, 7.43800790926512168161129403994, 8.651293583149554387589163788630, 10.06805256865888012741461433421, 11.18776118497176385692580865555, 13.11825660911732987690344773914, 14.52713725263599969640957445101, 15.37219563881707968698716748740