L(s) = 1 | + (0.258 + 0.965i)2-s + (−1.13 − 0.304i)3-s + (−0.866 + 0.499i)4-s − 1.17i·6-s + (−0.698 − 2.55i)7-s + (−0.707 − 0.707i)8-s + (−1.40 − 0.810i)9-s + (−0.371 − 0.643i)11-s + (1.13 − 0.304i)12-s + (2.05 − 2.05i)13-s + (2.28 − 1.33i)14-s + (0.500 − 0.866i)16-s + (1.69 − 6.33i)17-s + (0.419 − 1.56i)18-s + (−0.946 + 1.63i)19-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.655 − 0.175i)3-s + (−0.433 + 0.249i)4-s − 0.479i·6-s + (−0.264 − 0.964i)7-s + (−0.249 − 0.249i)8-s + (−0.467 − 0.270i)9-s + (−0.112 − 0.194i)11-s + (0.327 − 0.0877i)12-s + (0.570 − 0.570i)13-s + (0.610 − 0.356i)14-s + (0.125 − 0.216i)16-s + (0.411 − 1.53i)17-s + (0.0988 − 0.368i)18-s + (−0.217 + 0.375i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.656069 - 0.419790i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.656069 - 0.419790i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.698 + 2.55i)T \) |
good | 3 | \( 1 + (1.13 + 0.304i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (0.371 + 0.643i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.05 + 2.05i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.69 + 6.33i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.946 - 1.63i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.11 - 1.36i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 9.69iT - 29T^{2} \) |
| 31 | \( 1 + (-2.96 + 1.71i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.691 - 2.58i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 0.817iT - 41T^{2} \) |
| 43 | \( 1 + (1.59 + 1.59i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.54 - 1.21i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.29 - 4.81i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.27 - 2.20i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.25 - 3.03i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (13.2 + 3.54i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + (-8.54 - 2.29i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.70 - 3.29i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.23 + 9.23i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.01 - 5.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \) |
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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
−11.49076425570116698344692599822, −10.36888428013965575467507020682, −9.521577482092347384539195613214, −8.224296514388466645977298455935, −7.43980315071607289214372422865, −6.34256904996187912563372995392, −5.69401930744175028640768536617, −4.43586003585580542737566304042, −3.19205720911528063163332342319, −0.54910380064854861541863576838,
1.92904891740329212391392527082, 3.35949783084190610676858461640, 4.70321340966733271364425973192, 5.73178325174526342628729018418, 6.45647138635391451597995084507, 8.247694030407528488316657907370, 8.910187302377500907646292640611, 10.10851092623757838713441012007, 10.82112677566075630753487380707, 11.67145649078981581458741093690