| L(s) = 1 | + (−0.640 − 1.97i)2-s + (1.43 + 1.58i)3-s + (−1.85 + 1.34i)4-s + (2.21 − 3.83i)6-s + (−0.384 − 3.65i)7-s + (0.492 + 0.357i)8-s + (−0.164 + 1.56i)9-s + (−3.91 − 1.74i)11-s + (−4.79 − 1.02i)12-s + (−2.04 + 0.433i)13-s + (−6.95 + 3.09i)14-s + (−1.02 + 3.16i)16-s + (−1.94 + 0.865i)17-s + (3.19 − 0.679i)18-s + (0.606 + 0.128i)19-s + ⋯ |
| L(s) = 1 | + (−0.452 − 1.39i)2-s + (0.826 + 0.917i)3-s + (−0.927 + 0.674i)4-s + (0.904 − 1.56i)6-s + (−0.145 − 1.38i)7-s + (0.174 + 0.126i)8-s + (−0.0549 + 0.522i)9-s + (−1.17 − 0.524i)11-s + (−1.38 − 0.294i)12-s + (−0.566 + 0.120i)13-s + (−1.85 + 0.827i)14-s + (−0.256 + 0.790i)16-s + (−0.471 + 0.209i)17-s + (0.753 − 0.160i)18-s + (0.139 + 0.0295i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0570328 + 0.837687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0570328 + 0.837687i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + (1.44 + 5.37i)T \) |
| good | 2 | \( 1 + (0.640 + 1.97i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.43 - 1.58i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + (0.384 + 3.65i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.91 + 1.74i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (2.04 - 0.433i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (1.94 - 0.865i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.606 - 0.128i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (2.71 + 1.97i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.425 + 1.31i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.137 + 0.237i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.86 - 3.17i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.263 - 0.0560i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (1.66 - 5.11i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.993 + 9.45i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (3.89 + 4.33i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 2.22T + 61T^{2} \) |
| 67 | \( 1 + (6.80 + 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.139 + 1.32i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-12.9 - 5.76i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-7.92 + 3.52i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.46 + 3.85i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (4.05 - 2.94i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.43 + 3.94i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−9.890424042505644126407940630116, −9.501828587544249127128928271321, −8.415033370611455118011469917796, −7.73683359026266957679824455816, −6.41354305381987501416461983388, −4.77324019312647018963878488706, −3.92283683993950065415837906559, −3.20677172172708165820213526789, −2.20308672295119284292219422254, −0.40920651743630001280591879866,
2.14119279154239100761443953728, 2.88380163561568162827172932551, 4.99704196852892135163306986967, 5.62827914068093473577927263957, 6.71701500467655526613940126692, 7.42440837464551190591709039462, 8.041384103824137315142241295653, 8.756569183201398136979772352496, 9.346954380633975579606450784740, 10.42735431555909338713343046791
