L(s) = 1 | + (−0.309 + 1.37i)2-s + (−1.80 − 0.853i)4-s − i·5-s + (−2.64 − 0.0785i)7-s + (1.73 − 2.23i)8-s + (1.37 + 0.309i)10-s + 0.987i·11-s + 4.69i·13-s + (0.926 − 3.62i)14-s + (2.54 + 3.08i)16-s − 3.93i·17-s + 0.223·19-s + (−0.853 + 1.80i)20-s + (−1.36 − 0.305i)22-s − 5.88i·23-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.975i)2-s + (−0.904 − 0.426i)4-s − 0.447i·5-s + (−0.999 − 0.0296i)7-s + (0.614 − 0.789i)8-s + (0.436 + 0.0977i)10-s + 0.297i·11-s + 1.30i·13-s + (0.247 − 0.968i)14-s + (0.635 + 0.771i)16-s − 0.954i·17-s + 0.0513·19-s + (−0.190 + 0.404i)20-s + (−0.290 − 0.0650i)22-s − 1.22i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.097497493\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.097497493\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 1.37i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (2.64 + 0.0785i)T \) |
good | 11 | \( 1 - 0.987iT - 11T^{2} \) |
| 13 | \( 1 - 4.69iT - 13T^{2} \) |
| 17 | \( 1 + 3.93iT - 17T^{2} \) |
| 19 | \( 1 - 0.223T + 19T^{2} \) |
| 23 | \( 1 + 5.88iT - 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 - 4.32iT - 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 4.93iT - 61T^{2} \) |
| 67 | \( 1 - 7.84iT - 67T^{2} \) |
| 71 | \( 1 - 8.49iT - 71T^{2} \) |
| 73 | \( 1 + 14.4iT - 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 + 1.67T + 83T^{2} \) |
| 89 | \( 1 + 0.493iT - 89T^{2} \) |
| 97 | \( 1 + 4.31iT - 97T^{2} \) |
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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
−9.687777900414668571954577548875, −8.942582477750119429650074412105, −8.300229408615135335109752160807, −7.13130965716136100722398953706, −6.67760939000636187651667809965, −5.85366375513207632445887268887, −4.70660864201583521319829791840, −4.15750003156031251109409111149, −2.64920599915822795373162765010, −0.850378932465038820366381254308,
0.75622537465107746672268426215, 2.38088597606000106241309392159, 3.27905802540942943088971912699, 3.91211592227464564964414471780, 5.31842794955923246367085214521, 6.09088442991993420175533869200, 7.23885319101260471887743062310, 8.115977285915288794177083871979, 8.868918156623802613732667741924, 9.821225865726292650595643445700