L(s) = 1 | + (0.664 − 2.47i)2-s − 2.69i·3-s + (−3.97 − 2.29i)4-s + (0.103 + 0.384i)5-s + (−6.68 − 1.79i)6-s + (−4.70 + 4.70i)8-s − 4.27·9-s + 1.02·10-s + (2.56 − 2.56i)11-s + (−6.19 + 10.7i)12-s + (−3.44 + 1.06i)13-s + (1.03 − 0.277i)15-s + (3.95 + 6.84i)16-s + (−2.04 + 3.54i)17-s + (−2.83 + 10.5i)18-s + (0.569 − 0.569i)19-s + ⋯ |
L(s) = 1 | + (0.469 − 1.75i)2-s − 1.55i·3-s + (−1.98 − 1.14i)4-s + (0.0461 + 0.172i)5-s + (−2.72 − 0.731i)6-s + (−1.66 + 1.66i)8-s − 1.42·9-s + 0.323·10-s + (0.774 − 0.774i)11-s + (−1.78 + 3.09i)12-s + (−0.955 + 0.294i)13-s + (0.267 − 0.0717i)15-s + (0.987 + 1.71i)16-s + (−0.496 + 0.860i)17-s + (−0.668 + 2.49i)18-s + (0.130 − 0.130i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03262 + 0.901347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03262 + 0.901347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.44 - 1.06i)T \) |
good | 2 | \( 1 + (-0.664 + 2.47i)T + (-1.73 - i)T^{2} \) |
| 3 | \( 1 + 2.69iT - 3T^{2} \) |
| 5 | \( 1 + (-0.103 - 0.384i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.56 + 2.56i)T - 11iT^{2} \) |
| 17 | \( 1 + (2.04 - 3.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.569 + 0.569i)T - 19iT^{2} \) |
| 23 | \( 1 + (-4.41 + 2.54i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 1.74i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.06 + 1.62i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (1.73 + 0.463i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (0.578 + 2.15i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.19 + 2.19i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.54 + 7.87i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.17 + 1.92i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 2.77iT - 61T^{2} \) |
| 67 | \( 1 + (-3.55 - 3.55i)T + 67iT^{2} \) |
| 71 | \( 1 + (0.582 - 2.17i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.28 + 4.78i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.80 + 3.13i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.36 - 5.36i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.09 - 11.5i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-7.71 - 2.06i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.23747708379045968636091763805, −9.113424328371819586989315107190, −8.444991005706306537786295872824, −7.11225857107440327458960437390, −6.28640453658847923206046020400, −5.10008427159374015897416069117, −3.84732166593235055036799375147, −2.68444003557805354482941747810, −1.84690906328714553594185212929, −0.65901264409958786112845003667,
3.23509365226687853149762015643, 4.32940509811098068427145598738, 4.93160894142671197931751290093, 5.53616993113629560825141290201, 6.89085949184419067006516163015, 7.43770111743984461290165272430, 8.857581522077836264538653275312, 9.184379909166984199573927961296, 9.979351387838336375771745559464, 11.10885844003412383544548265185