L(s) = 1 | + (0.554 − 2.06i)2-s + 0.0197i·3-s + (−2.24 − 1.29i)4-s + (0.360 + 1.34i)5-s + (0.0408 + 0.0109i)6-s + (−0.893 + 0.893i)8-s + 2.99·9-s + 2.98·10-s + (−0.246 + 0.246i)11-s + (0.0255 − 0.0442i)12-s + (−1.32 − 3.35i)13-s + (−0.0265 + 0.00711i)15-s + (−1.23 − 2.14i)16-s + (0.491 − 0.850i)17-s + (1.66 − 6.20i)18-s + (3.25 − 3.25i)19-s + ⋯ |
L(s) = 1 | + (0.392 − 1.46i)2-s + 0.0113i·3-s + (−1.12 − 0.647i)4-s + (0.161 + 0.601i)5-s + (0.0166 + 0.00446i)6-s + (−0.315 + 0.315i)8-s + 0.999·9-s + 0.943·10-s + (−0.0743 + 0.0743i)11-s + (0.00737 − 0.0127i)12-s + (−0.368 − 0.929i)13-s + (−0.00685 + 0.00183i)15-s + (−0.309 − 0.535i)16-s + (0.119 − 0.206i)17-s + (0.392 − 1.46i)18-s + (0.747 − 0.747i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.924395 - 1.71209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.924395 - 1.71209i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (1.32 + 3.35i)T \) |
good | 2 | \( 1 + (-0.554 + 2.06i)T + (-1.73 - i)T^{2} \) |
| 3 | \( 1 - 0.0197iT - 3T^{2} \) |
| 5 | \( 1 + (-0.360 - 1.34i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (0.246 - 0.246i)T - 11iT^{2} \) |
| 17 | \( 1 + (-0.491 + 0.850i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.25 + 3.25i)T - 19iT^{2} \) |
| 23 | \( 1 + (-2.86 + 1.65i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.941 - 1.63i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.81 - 0.755i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (7.91 + 2.12i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.580 - 2.16i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (6.47 - 3.73i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.5 + 2.83i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.77 - 6.53i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (14.7 - 3.94i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 6.45iT - 61T^{2} \) |
| 67 | \( 1 + (-7.13 - 7.13i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.43 + 9.07i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (2.76 - 10.3i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.890 + 1.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.33 + 8.33i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.51 - 13.1i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (12.6 + 3.39i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.45996031287226031879054989340, −9.865625441232546429543322589343, −8.941574896045652390747774429903, −7.49149627106687372271358480172, −6.83172061563693093876453194549, −5.30675488063260539359080209947, −4.46905281303080451257106246237, −3.27087310423374843317713895754, −2.53489207864306739551406801745, −1.08736122411522090770657638511,
1.65653089057888374275232569904, 3.74681335819942387628584874835, 4.76643021732333730849648484265, 5.39359041796029878932709231509, 6.53095166751474506384689171477, 7.19614744498124585990439445670, 7.995703891212363047280485794612, 8.955142958197618984110801911217, 9.700418643137504919041144991547, 10.77608129397622435971805502814