L(s) = 1 | + (−1.13 − 0.847i)2-s + (−1.71 − 0.242i)3-s + (0.562 + 1.91i)4-s + (1.73 + 1.72i)6-s + 3.08·7-s + (0.990 − 2.64i)8-s + (2.88 + 0.831i)9-s + 2.54i·11-s + (−0.499 − 3.42i)12-s − 5.06·13-s + (−3.49 − 2.61i)14-s + (−3.36 + 2.15i)16-s − 0.214·17-s + (−2.55 − 3.38i)18-s − 2.60·19-s + ⋯ |
L(s) = 1 | + (−0.800 − 0.599i)2-s + (−0.990 − 0.139i)3-s + (0.281 + 0.959i)4-s + (0.708 + 0.705i)6-s + 1.16·7-s + (0.350 − 0.936i)8-s + (0.960 + 0.277i)9-s + 0.767i·11-s + (−0.144 − 0.989i)12-s − 1.40·13-s + (−0.934 − 0.700i)14-s + (−0.841 + 0.539i)16-s − 0.0519·17-s + (−0.602 − 0.797i)18-s − 0.598·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.627213 + 0.197089i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.627213 + 0.197089i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.13 + 0.847i)T \) |
| 3 | \( 1 + (1.71 + 0.242i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 5.06T + 13T^{2} \) |
| 17 | \( 1 + 0.214T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 - 4.58iT - 31T^{2} \) |
| 37 | \( 1 - 7.67T + 37T^{2} \) |
| 41 | \( 1 + 9.26iT - 41T^{2} \) |
| 43 | \( 1 - 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 9.51iT - 53T^{2} \) |
| 59 | \( 1 - 0.428iT - 59T^{2} \) |
| 61 | \( 1 - 1.11iT - 61T^{2} \) |
| 67 | \( 1 + 2.35iT - 67T^{2} \) |
| 71 | \( 1 - 6.12T + 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 12.4iT - 89T^{2} \) |
| 97 | \( 1 - 8.04iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80027897034078917108295478419, −10.03082809412475139406495976902, −9.251207571205066978661109435112, −7.924722229544011853021067918711, −7.45840163972683895063962220055, −6.44177647111087053106327311658, −4.96223697865605581946520873215, −4.37020090773701875373938482883, −2.45934398120008070154685755951, −1.30395055474248586566190854379,
0.59954649658720931841777104551, 2.18903777062110585274437931907, 4.51015576753234986868202954738, 5.13192280036220605917093854723, 6.14795364232548152957558996085, 6.98956828933238965812738217744, 7.953989143758786927198505890648, 8.677943408244848494656594196823, 9.903982478334906343629362164987, 10.43173794836626094644403478074