| L(s) = 1 | + (1.32 + 0.484i)2-s + (0.943 − 0.236i)3-s + (1.52 + 1.28i)4-s + (1.14 + 0.542i)5-s + (1.36 + 0.143i)6-s + (−1.25 − 4.14i)7-s + (1.40 + 2.45i)8-s + (−1.81 + 0.967i)9-s + (1.26 + 1.27i)10-s + (0.162 − 0.0240i)11-s + (1.74 + 0.854i)12-s + (−0.288 + 0.807i)13-s + (0.340 − 6.12i)14-s + (1.21 + 0.241i)15-s + (0.679 + 3.94i)16-s + (−5.47 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.544 − 0.136i)3-s + (0.764 + 0.644i)4-s + (0.513 + 0.242i)5-s + (0.558 + 0.0586i)6-s + (−0.475 − 1.56i)7-s + (0.497 + 0.867i)8-s + (−0.603 + 0.322i)9-s + (0.398 + 0.404i)10-s + (0.0489 − 0.00725i)11-s + (0.504 + 0.246i)12-s + (−0.0801 + 0.223i)13-s + (0.0908 − 1.63i)14-s + (0.312 + 0.0622i)15-s + (0.169 + 0.985i)16-s + (−1.32 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37909 + 0.440150i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37909 + 0.440150i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 - 0.484i)T \) |
| good | 3 | \( 1 + (-0.943 + 0.236i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-1.14 - 0.542i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (1.25 + 4.14i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-0.162 + 0.0240i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (0.288 - 0.807i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (5.47 - 1.08i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (0.0753 + 0.0683i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.126 + 0.0124i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-3.26 - 4.40i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-5.88 - 2.43i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.493 + 10.0i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-1.36 + 1.66i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-1.43 + 5.72i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-1.33 - 0.890i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (6.54 - 8.81i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (1.40 + 3.92i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-1.42 + 2.37i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (5.65 + 3.38i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-7.00 + 13.1i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (2.14 - 7.06i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-6.68 - 10.0i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.311 + 6.34i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-12.0 - 1.18i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (5.60 - 13.5i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−12.39239017243616198914057628434, −11.01945204689204139547283368290, −10.48944453132286911964664659621, −9.050933840590687217202070640658, −7.87741325709604930447804568560, −6.95530660822726005725371587646, −6.15675585780937663958253645934, −4.66058586322244245289415261032, −3.58957972909961858960944299624, −2.30429646394817517236767643487,
2.27969589161482766542441994441, 3.06627093024277552168450732887, 4.67073815873904101791657589220, 5.84139163311516371557006083458, 6.45274783705245189756377474372, 8.265471832275397484510165211659, 9.248985377087902868189363629557, 9.911204130414973906609004282219, 11.43869068461933658544187727787, 11.94332703106391843886231807039
