| L(s) = 1 | + (−0.0782 − 1.41i)2-s + (0.746 + 1.56i)3-s + (−1.98 + 0.221i)4-s + (−0.241 + 0.820i)5-s + (2.14 − 1.17i)6-s + (2.26 + 1.96i)7-s + (0.467 + 2.78i)8-s + (−1.88 + 2.33i)9-s + (1.17 + 0.276i)10-s + (−5.20 + 3.34i)11-s + (−1.82 − 2.94i)12-s + (−0.429 − 0.495i)13-s + (2.59 − 3.35i)14-s + (−1.46 + 0.235i)15-s + (3.90 − 0.879i)16-s + (0.947 + 0.432i)17-s + ⋯ |
| L(s) = 1 | + (−0.0553 − 0.998i)2-s + (0.430 + 0.902i)3-s + (−0.993 + 0.110i)4-s + (−0.107 + 0.367i)5-s + (0.877 − 0.480i)6-s + (0.857 + 0.743i)7-s + (0.165 + 0.986i)8-s + (−0.628 + 0.777i)9-s + (0.372 + 0.0873i)10-s + (−1.56 + 1.00i)11-s + (−0.527 − 0.849i)12-s + (−0.119 − 0.137i)13-s + (0.694 − 0.897i)14-s + (−0.377 + 0.0608i)15-s + (0.975 − 0.219i)16-s + (0.229 + 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14538 + 0.407759i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14538 + 0.407759i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0782 + 1.41i)T \) |
| 3 | \( 1 + (-0.746 - 1.56i)T \) |
| 23 | \( 1 + (-0.944 + 4.70i)T \) |
| good | 5 | \( 1 + (0.241 - 0.820i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-2.26 - 1.96i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (5.20 - 3.34i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.429 + 0.495i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.947 - 0.432i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-6.17 + 2.82i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-8.05 - 3.67i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (3.76 + 0.540i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.27 - 0.668i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.65 + 5.62i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (5.95 - 0.855i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 + (2.19 + 1.89i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (5.85 + 6.75i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.35 - 9.45i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.75 + 5.85i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.71 - 3.67i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.30 + 2.85i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.12 + 7.04i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.62 + 1.65i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-4.31 + 0.620i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.80 + 1.11i)T + (81.6 + 52.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.84148298708147815277641045449, −10.79604241277475957258650287734, −10.33836032032512455341542138519, −9.304655707994229284047545339841, −8.420273756704868689231571484514, −7.53441020261198861163364233295, −5.10673288729716340101145641646, −4.93311612934403675108440414879, −3.17729886435238320116641607746, −2.30984584188952982525090047046,
0.985144931617090806792326471936, 3.26217744714519845965773632935, 4.89818937234427145897611682747, 5.80059868097742916728620748912, 7.15524032560075907371564195755, 7.952060274357077027750881914547, 8.312861357562052513046789846511, 9.606935039580235810990006369568, 10.79672680355420955074248682148, 12.01574731307402108381177480260
