| L(s) = 1 | + (−1.27 + 0.618i)2-s + (−0.216 − 1.08i)3-s + (1.23 − 1.57i)4-s + (1.50 − 1.00i)5-s + (0.947 + 1.24i)6-s + (1.15 + 0.477i)7-s + (−0.595 + 2.76i)8-s + (1.63 − 0.678i)9-s + (−1.28 + 2.20i)10-s + (−4.61 − 0.918i)11-s + (−1.97 − 1.00i)12-s + (1.18 + 0.790i)13-s + (−1.76 + 0.106i)14-s + (−1.41 − 1.41i)15-s + (−0.954 − 3.88i)16-s + (−4.96 + 4.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.899 + 0.437i)2-s + (−0.124 − 0.627i)3-s + (0.616 − 0.786i)4-s + (0.671 − 0.448i)5-s + (0.386 + 0.509i)6-s + (0.435 + 0.180i)7-s + (−0.210 + 0.977i)8-s + (0.545 − 0.226i)9-s + (−0.407 + 0.697i)10-s + (−1.39 − 0.276i)11-s + (−0.570 − 0.288i)12-s + (0.328 + 0.219i)13-s + (−0.470 + 0.0284i)14-s + (−0.365 − 0.365i)15-s + (−0.238 − 0.971i)16-s + (−1.20 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.654739 - 0.107607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.654739 - 0.107607i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 - 0.618i)T \) |
| good | 3 | \( 1 + (0.216 + 1.08i)T + (-2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (-1.50 + 1.00i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.15 - 0.477i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (4.61 + 0.918i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-1.18 - 0.790i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (4.96 - 4.96i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.306 + 0.458i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-2.28 - 5.52i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-4.74 + 0.944i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 + (1.95 + 2.92i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (2.30 + 5.56i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (1.76 - 8.86i)T + (-39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-6.66 + 6.66i)T - 47iT^{2} \) |
| 53 | \( 1 + (13.5 + 2.68i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-4.57 + 3.05i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (-1.79 - 9.02i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (0.821 + 4.12i)T + (-61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 4.64i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-7.67 + 3.17i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (7.40 + 7.40i)T + 79iT^{2} \) |
| 83 | \( 1 + (-3.58 + 5.36i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-0.0401 + 0.0970i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 7.44iT - 97T^{2} \) |
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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
−15.25086705272127775460228303207, −13.67189815603262904719472721251, −12.83130335233546180311303486156, −11.28775254222216588494619644374, −10.19448893948566487445757247982, −8.911155022551748108381478149485, −7.85350442806561113994736219714, −6.53308666642934199718399248517, −5.28438779454400683123725485922, −1.79855874547266300301774025042,
2.53708327852582823382581373082, 4.75162868486532092982246963424, 6.78479377910797611100850048676, 8.105384445915017319617512670595, 9.546980349811295074119810204033, 10.46294354369955653194777657778, 11.05801607622621013013249372085, 12.69406348714163494454746562797, 13.77846247684238540524181709688, 15.45824072204199890239506868831
