| L(s) = 1 | + (−0.160 − 0.0140i)2-s + (−1.94 − 0.342i)4-s + (0.476 + 2.18i)5-s + (−2.37 − 1.66i)7-s + (0.617 + 0.165i)8-s + (−0.0456 − 0.356i)10-s + (1.48 − 4.09i)11-s + (−0.412 − 4.71i)13-s + (0.356 + 0.299i)14-s + (3.61 + 1.31i)16-s + (5.66 − 1.51i)17-s + (0.695 − 0.401i)19-s + (−0.176 − 4.41i)20-s + (−0.296 + 0.635i)22-s + (−1.52 − 2.18i)23-s + ⋯ |
| L(s) = 1 | + (−0.113 − 0.00991i)2-s + (−0.972 − 0.171i)4-s + (0.212 + 0.977i)5-s + (−0.896 − 0.627i)7-s + (0.218 + 0.0585i)8-s + (−0.0144 − 0.112i)10-s + (0.449 − 1.23i)11-s + (−0.114 − 1.30i)13-s + (0.0953 + 0.0800i)14-s + (0.903 + 0.328i)16-s + (1.37 − 0.368i)17-s + (0.159 − 0.0920i)19-s + (−0.0394 − 0.986i)20-s + (−0.0631 + 0.135i)22-s + (−0.318 − 0.455i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.646582 - 0.517042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.646582 - 0.517042i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.476 - 2.18i)T \) |
| good | 2 | \( 1 + (0.160 + 0.0140i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (2.37 + 1.66i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.48 + 4.09i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.412 + 4.71i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-5.66 + 1.51i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.695 + 0.401i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.52 + 2.18i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (2.06 - 1.73i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 7.23i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.841 + 3.14i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.88 - 3.44i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.90 + 6.22i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (3.17 - 4.52i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.70 + 1.70i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.31 + 1.57i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.10 - 11.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.65 - 0.582i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-5.61 - 3.24i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.49 + 9.31i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.322 + 0.384i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.780 + 8.92i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-6.84 - 11.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.60 + 3.08i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.73630968574037110066622498677, −10.12423858420807865382457283738, −9.490291589240479603252970069403, −8.274718093115253550453149775014, −7.41046112467796865700899710586, −6.17010048084521429011255991933, −5.45603086279410905249743314479, −3.75115298587931973406005895761, −3.12141847978682974822897515253, −0.61305738115205673928502334144,
1.61813760210631701019077345371, 3.58288404823801596334033927082, 4.62323235853229240098812607293, 5.51729514472619127944987085046, 6.73017359213762821272766035573, 7.979414662881680603623343267765, 8.920139644466763970508939451887, 9.625542388432066927957824214399, 9.961123197701585678833010629423, 11.92907520047227523909770461918
