| L(s) = 1 | + (0.0403 + 0.254i)2-s + (−1.86 − 0.952i)3-s + (1.83 − 0.597i)4-s + (−1.59 + 1.56i)5-s + (0.167 − 0.514i)6-s + (−0.170 + 0.170i)7-s + (0.460 + 0.903i)8-s + (0.821 + 1.13i)9-s + (−0.463 − 0.343i)10-s + (2.38 + 1.73i)11-s + (−4.00 − 0.634i)12-s + (−0.220 + 1.39i)13-s + (−0.0503 − 0.0365i)14-s + (4.47 − 1.40i)15-s + (2.91 − 2.11i)16-s + (1.11 + 2.18i)17-s + ⋯ |
| L(s) = 1 | + (0.0285 + 0.180i)2-s + (−1.07 − 0.549i)3-s + (0.919 − 0.298i)4-s + (−0.713 + 0.700i)5-s + (0.0682 − 0.209i)6-s + (−0.0645 + 0.0645i)7-s + (0.162 + 0.319i)8-s + (0.273 + 0.376i)9-s + (−0.146 − 0.108i)10-s + (0.717 + 0.521i)11-s + (−1.15 − 0.183i)12-s + (−0.0611 + 0.386i)13-s + (−0.0134 − 0.00977i)14-s + (1.15 − 0.363i)15-s + (0.729 − 0.529i)16-s + (0.269 + 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08739 + 0.278299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08739 + 0.278299i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.59 - 1.56i)T \) |
| 19 | \( 1 + (-4.16 - 1.27i)T \) |
| good | 2 | \( 1 + (-0.0403 - 0.254i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (1.86 + 0.952i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (0.170 - 0.170i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.38 - 1.73i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.220 - 1.39i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.11 - 2.18i)T + (-9.99 + 13.7i)T^{2} \) |
| 23 | \( 1 + (-0.469 - 2.96i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (2.44 + 7.53i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.65 - 2.16i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-10.7 - 1.70i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.75 - 7.92i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (4.80 + 4.80i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.347 + 0.177i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-2.66 - 1.35i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (8.85 - 6.43i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.61 + 1.17i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (9.29 - 4.73i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (4.61 - 1.49i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (2.44 + 15.4i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.25 + 6.93i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (8.96 - 4.56i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.37 - 6.08i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.44 + 8.72i)T + (-57.0 - 78.4i)T^{2} \) |
| 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
−11.41146176483226365021402475112, −10.45482754996229085806941743088, −9.530829766342409128604615206237, −7.84367117016955552927366109325, −7.28677430343486079591813619602, −6.32870816454149852794591611048, −5.93510139461905793992067724633, −4.42827084594948438394358902389, −2.95706538676650237550278075665, −1.32952135993418258842975255197,
0.913581731177868750855892983894, 3.03247298168663144599725899136, 4.19194980448566237302086264238, 5.23016411984807467871451148172, 6.16121250028995277260118227850, 7.22533456712663410601707326501, 8.132940667996664871204212106066, 9.277758951615979430649207811834, 10.33196548444529944114574243708, 11.23694626200837051793548983102
