L(s) = 1 | + (−1.40 − 0.159i)2-s + 3-s + (1.94 + 0.449i)4-s + 0.947i·5-s + (−1.40 − 0.159i)6-s − 0.0251·7-s + (−2.66 − 0.943i)8-s + 9-s + (0.151 − 1.33i)10-s − 1.77·11-s + (1.94 + 0.449i)12-s − 5.89i·13-s + (0.0352 + 0.00401i)14-s + 0.947i·15-s + (3.59 + 1.75i)16-s + 2.08·17-s + ⋯ |
L(s) = 1 | + (−0.993 − 0.113i)2-s + 0.577·3-s + (0.974 + 0.224i)4-s + 0.423i·5-s + (−0.573 − 0.0652i)6-s − 0.00949·7-s + (−0.942 − 0.333i)8-s + 0.333·9-s + (0.0478 − 0.420i)10-s − 0.536·11-s + (0.562 + 0.129i)12-s − 1.63i·13-s + (0.00943 + 0.00107i)14-s + 0.244i·15-s + (0.899 + 0.437i)16-s + 0.505·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12857 - 0.378227i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12857 - 0.378227i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.159i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (-7.47 - 3.33i)T \) |
good | 5 | \( 1 - 0.947iT - 5T^{2} \) |
| 7 | \( 1 + 0.0251T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 + 5.89iT - 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 + 4.96iT - 19T^{2} \) |
| 23 | \( 1 + 4.05iT - 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 - 5.82T + 37T^{2} \) |
| 41 | \( 1 - 6.60iT - 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 + 7.44iT - 53T^{2} \) |
| 59 | \( 1 - 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 2.42iT - 61T^{2} \) |
| 71 | \( 1 + 1.82iT - 71T^{2} \) |
| 73 | \( 1 + 9.08T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 9.99T + 89T^{2} \) |
| 97 | \( 1 + 3.04iT - 97T^{2} \) |
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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
−10.20158347073389471751066826490, −9.362863490967132124159948083407, −8.310579801898497684559346091778, −7.928308547386166816626362722579, −6.95991284984453724541003594115, −6.07421467723131204008762222768, −4.75827609942916303742057324716, −2.99845729539994135341945962623, −2.73842726617020628679025995683, −0.863539949730038268369293608585,
1.32308786165168914760434611037, 2.44875032074281081561618137515, 3.75415904160156714525309028061, 5.06310728234257693033095444550, 6.24688640246122683569351110139, 7.12099207625913073650103868070, 8.005741270192064601109159803503, 8.634446171600377293282008295853, 9.460884182651311527201201260168, 10.04234449617599863619075841960