L(s) = 1 | + i·2-s − 3-s − 4-s − 2i·5-s − i·6-s − i·8-s + 9-s + 2·10-s + 12-s + (3 + 2i)13-s + 2i·15-s + 16-s + 2·17-s + i·18-s + 6i·19-s + 2i·20-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.894i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s + 0.632·10-s + 0.288·12-s + (0.832 + 0.554i)13-s + 0.516i·15-s + 0.250·16-s + 0.485·17-s + 0.235i·18-s + 1.37i·19-s + 0.447i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.408352940\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.408352940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-3 - 2i)T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.508678442950309649005647619017, −7.83144886322286838288380768687, −7.18606805449285450629387747142, −6.11725242134247229214825955898, −5.81251956109654462293586506546, −4.97300766978274485378060510492, −4.21043966905202273620386678418, −3.48681924873775414111823132062, −1.79502897079381486437847821399, −0.844615223775748613531646780089,
0.63532087720513272905995758472, 1.79593973989094017552982170993, 3.01230306631566244689649768439, 3.43753072507114542699678702954, 4.57151468145937373871369189002, 5.34711438141615528898344326510, 6.06845525461585603333445897033, 6.99727455278791735341603897689, 7.43831754561200452866466859406, 8.592443095006908719630686474791