| L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.104 − 0.994i)3-s + (0.309 − 0.951i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (3.97 − 0.844i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (0.847 + 0.940i)11-s + (−0.913 − 0.406i)12-s + (3.89 − 1.73i)13-s + (−2.71 + 3.01i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−2.30 + 2.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.0603 − 0.574i)3-s + (0.154 − 0.475i)4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (1.50 − 0.319i)7-s + (0.109 + 0.336i)8-s + (−0.326 − 0.0693i)9-s + (−0.0330 − 0.314i)10-s + (0.255 + 0.283i)11-s + (−0.263 − 0.117i)12-s + (1.08 − 0.481i)13-s + (−0.726 + 0.806i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (−0.558 + 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43705 - 0.0376315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43705 - 0.0376315i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-2.52 - 4.96i)T \) |
| good | 7 | \( 1 + (-3.97 + 0.844i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (-0.847 - 0.940i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.89 + 1.73i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (2.30 - 2.55i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.106 - 0.0475i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.0893 + 0.275i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.57 + 1.14i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-0.591 - 1.02i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.840 + 7.99i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (5.44 + 2.42i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-6.60 - 4.80i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.08 - 1.29i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.472 + 4.49i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + (0.0802 - 0.138i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.25 - 0.266i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (3.86 + 4.28i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-6.35 + 7.05i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.308 + 2.93i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-2.66 + 8.21i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (3.84 - 11.8i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.22683134971300109635914577447, −8.774881128062301521146258945368, −8.388103638718442420216630795081, −7.60159535412514219021483326193, −6.84879477287776449336357492255, −5.96042617049146344464014381788, −4.90123594598088093790156899296, −3.76405410248036365974461929796, −2.14870637031588973999859105645, −1.09789095990601636098183102138,
1.16092408246547392140778125060, 2.39936298327138451958404404937, 3.82945275050318960977768431323, 4.59817430063268028990576396854, 5.55404775922335158738639697364, 6.78803263182959112658037320250, 8.008720133236738447015208159097, 8.508815962195915765485848852911, 9.111553626216207562912865660134, 10.04662653189704391299185708350
