| L(s) = 1 | + (1.92 + 1.73i)2-s + (−0.994 + 0.104i)3-s + (0.493 + 4.69i)4-s + (1.13 + 1.92i)5-s + (−2.09 − 1.52i)6-s + (2.57 − 0.590i)7-s + (−4.14 + 5.69i)8-s + (0.978 − 0.207i)9-s + (−1.15 + 5.67i)10-s + (−1.36 − 0.290i)11-s + (−0.980 − 4.61i)12-s + (1.76 + 0.572i)13-s + (5.99 + 3.33i)14-s + (−1.33 − 1.79i)15-s + (−8.62 + 1.83i)16-s + (−2.96 − 6.66i)17-s + ⋯ |
| L(s) = 1 | + (1.36 + 1.22i)2-s + (−0.574 + 0.0603i)3-s + (0.246 + 2.34i)4-s + (0.507 + 0.861i)5-s + (−0.856 − 0.621i)6-s + (0.974 − 0.223i)7-s + (−1.46 + 2.01i)8-s + (0.326 − 0.0693i)9-s + (−0.365 + 1.79i)10-s + (−0.412 − 0.0876i)11-s + (−0.283 − 1.33i)12-s + (0.488 + 0.158i)13-s + (1.60 + 0.891i)14-s + (−0.343 − 0.464i)15-s + (−2.15 + 0.458i)16-s + (−0.719 − 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.955417 + 2.61481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.955417 + 2.61481i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.994 - 0.104i)T \) |
| 5 | \( 1 + (-1.13 - 1.92i)T \) |
| 7 | \( 1 + (-2.57 + 0.590i)T \) |
| good | 2 | \( 1 + (-1.92 - 1.73i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (1.36 + 0.290i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.76 - 0.572i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.96 + 6.66i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.296 + 2.81i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.912 + 0.821i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (3.85 - 2.79i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.38 - 0.614i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.77 - 8.33i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.83 + 8.72i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.56iT - 43T^{2} \) |
| 47 | \( 1 + (-5.35 + 12.0i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.24 + 0.235i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (3.64 + 4.04i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.959 + 1.06i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (3.24 + 7.28i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-12.8 + 9.36i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.65 - 12.4i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-8.98 - 3.99i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (7.98 - 10.9i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (6.19 - 6.88i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-8.22 - 11.3i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.33607908410131075266177827445, −10.73265421824514181006807192410, −9.269640094909544811561330844540, −8.046819145473415274331040124417, −7.08806464152210456728786500704, −6.66485581745536113786244192011, −5.46801977440484827518486229924, −4.98472194755853757272927422813, −3.84229039229603645287809857138, −2.50709277849811948164120738071,
1.35615515699133918248908645037, 2.20008520139908664732296629124, 3.99298929133030723573080590434, 4.61798705542662043291957252545, 5.77503534400937244568185453139, 5.93561269180010887558308893697, 7.86796056217633405268742966010, 9.011808462595623197481136108320, 10.12487701504638285295763733495, 10.86028368317040018633336571339
