| L(s) = 1 | + (0.140 + 0.101i)2-s + (−0.309 − 0.951i)3-s + (−0.608 − 1.87i)4-s + (−0.103 + 2.23i)5-s + (0.0535 − 0.164i)6-s − 7-s + (0.212 − 0.654i)8-s + (−0.809 + 0.587i)9-s + (−0.242 + 0.302i)10-s + (−3.98 − 2.89i)11-s + (−1.59 + 1.15i)12-s + (−3.12 + 2.26i)13-s + (−0.140 − 0.101i)14-s + (2.15 − 0.591i)15-s + (−3.09 + 2.24i)16-s + (1.66 − 5.13i)17-s + ⋯ |
| L(s) = 1 | + (0.0991 + 0.0720i)2-s + (−0.178 − 0.549i)3-s + (−0.304 − 0.936i)4-s + (−0.0462 + 0.998i)5-s + (0.0218 − 0.0673i)6-s − 0.377·7-s + (0.0751 − 0.231i)8-s + (−0.269 + 0.195i)9-s + (−0.0765 + 0.0957i)10-s + (−1.20 − 0.872i)11-s + (−0.460 + 0.334i)12-s + (−0.866 + 0.629i)13-s + (−0.0374 − 0.0272i)14-s + (0.556 − 0.152i)15-s + (−0.772 + 0.561i)16-s + (0.404 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0203373 + 0.185653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0203373 + 0.185653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.103 - 2.23i)T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + (-0.140 - 0.101i)T + (0.618 + 1.90i)T^{2} \) |
| 11 | \( 1 + (3.98 + 2.89i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (3.12 - 2.26i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.66 + 5.13i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.78 - 5.49i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.272 + 0.197i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.417 + 1.28i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.343 - 1.05i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.76 + 2.73i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (8.43 - 6.12i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + (2.99 + 9.20i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.75 - 5.38i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.51 + 5.46i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-11.2 - 8.20i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.34 + 4.13i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.22 + 6.84i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.13 + 2.27i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.05 + 6.31i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.87 + 11.9i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.344 + 0.250i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.998 + 3.07i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.14511940660881983701173599374, −9.959014158950165220166963585953, −8.525391676625142746833481027839, −7.47618910110525736718642773578, −6.64934677105441033089594263118, −5.81516697282312164955187257597, −4.94859981616864783069253463368, −3.32417886631998740898662537605, −2.11953638421120297921445598996, −0.10065117824282118031193462939,
2.46490449538956309088212523810, 3.73289280150574566033027953434, 4.79755580672575711210574743755, 5.31326777827316544470996116676, 6.92661742899860830166283786946, 8.009526640368887520975311727559, 8.550109336645191045466502980134, 9.669886740058213179229754377557, 10.23608679589131765242568441260, 11.45841659366183355369390255030
