| L(s) = 1 | + (−0.804 + 0.593i)2-s + (−0.162 + 0.858i)3-s + (0.294 − 0.955i)4-s + (2.21 − 0.276i)5-s + (−0.379 − 0.787i)6-s + (0.601 − 2.57i)7-s + (0.330 + 0.943i)8-s + (2.08 + 0.816i)9-s + (−1.62 + 1.53i)10-s + (−2.11 − 5.38i)11-s + (0.772 + 0.408i)12-s + (0.0911 + 0.808i)13-s + (1.04 + 2.43i)14-s + (−0.123 + 1.95i)15-s + (−0.826 − 0.563i)16-s + (−0.128 − 3.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.568 + 0.419i)2-s + (−0.0938 + 0.495i)3-s + (0.147 − 0.477i)4-s + (0.992 − 0.123i)5-s + (−0.154 − 0.321i)6-s + (0.227 − 0.973i)7-s + (0.116 + 0.333i)8-s + (0.693 + 0.272i)9-s + (−0.512 + 0.486i)10-s + (−0.637 − 1.62i)11-s + (0.223 + 0.117i)12-s + (0.0252 + 0.224i)13-s + (0.279 + 0.649i)14-s + (−0.0318 + 0.503i)15-s + (−0.206 − 0.140i)16-s + (−0.0311 − 0.831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23044 - 0.143143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23044 - 0.143143i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.804 - 0.593i)T \) |
| 5 | \( 1 + (-2.21 + 0.276i)T \) |
| 7 | \( 1 + (-0.601 + 2.57i)T \) |
| good | 3 | \( 1 + (0.162 - 0.858i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (2.11 + 5.38i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.0911 - 0.808i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (0.128 + 3.43i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (3.76 + 6.51i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.579 + 0.0216i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-0.638 - 0.145i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.49 - 2.59i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.66 - 8.82i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 2.71i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-8.71 - 3.04i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-5.19 - 7.03i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-0.271 - 0.513i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.490 + 6.54i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-3.33 - 10.8i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-2.33 - 8.69i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.12 + 4.91i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.801 - 1.08i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-7.93 + 4.58i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.96 + 1.01i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (5.80 - 14.7i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-2.71 - 2.71i)T + 97iT^{2} \) |
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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
−10.70612042988863535434976768506, −10.09249039876988053403567949787, −9.163157372944560119279167850738, −8.379157301936197290965985065182, −7.23284576294375293717048856646, −6.41101486296838313785676121900, −5.27654538837479361478988776326, −4.44762131880360866968299821477, −2.73686402118672153051660801585, −0.978411328685979069975167324872,
1.77310222240557633662114394235, 2.29548212976790723846192367944, 4.14839486904932749791354820132, 5.53302281669268875605596848405, 6.43224859254722124730888371418, 7.45924353962836512537524724277, 8.349871611556412782723131104617, 9.409793822827272184307795116388, 10.10115353098389688971392028930, 10.68271254505893256650099432284
