| L(s) = 1 | + (−1.26 + 0.629i)2-s + (1.72 + 0.165i)3-s + (1.20 − 1.59i)4-s + (−1.13 − 0.558i)5-s + (−2.28 + 0.875i)6-s + (3.03 + 3.95i)7-s + (−0.524 + 2.77i)8-s + (2.94 + 0.571i)9-s + (1.78 − 0.00593i)10-s + (1.97 + 1.72i)11-s + (2.34 − 2.54i)12-s + (−0.913 − 0.0598i)13-s + (−6.34 − 3.10i)14-s + (−1.85 − 1.15i)15-s + (−1.08 − 3.84i)16-s + (−3.51 − 3.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.895 + 0.445i)2-s + (0.995 + 0.0957i)3-s + (0.603 − 0.797i)4-s + (−0.506 − 0.249i)5-s + (−0.933 + 0.357i)6-s + (1.14 + 1.49i)7-s + (−0.185 + 0.982i)8-s + (0.981 + 0.190i)9-s + (0.564 − 0.00187i)10-s + (0.594 + 0.521i)11-s + (0.677 − 0.735i)12-s + (−0.253 − 0.0165i)13-s + (−1.69 − 0.828i)14-s + (−0.480 − 0.296i)15-s + (−0.271 − 0.962i)16-s + (−0.852 − 0.852i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17816 + 0.798893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17816 + 0.798893i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.629i)T \) |
| 3 | \( 1 + (-1.72 - 0.165i)T \) |
| good | 5 | \( 1 + (1.13 + 0.558i)T + (3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (-3.03 - 3.95i)T + (-1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (-1.97 - 1.72i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.913 + 0.0598i)T + (12.8 + 1.69i)T^{2} \) |
| 17 | \( 1 + (3.51 + 3.51i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.121 + 0.182i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (2.61 - 3.40i)T + (-5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (0.276 - 0.815i)T + (-23.0 - 17.6i)T^{2} \) |
| 31 | \( 1 + (-5.21 - 9.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.70 + 3.14i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (4.50 + 3.45i)T + (10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (-1.18 - 1.03i)T + (5.61 + 42.6i)T^{2} \) |
| 47 | \( 1 + (-0.499 - 1.86i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-12.9 - 2.58i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-2.79 + 5.67i)T + (-35.9 - 46.8i)T^{2} \) |
| 61 | \( 1 + (-1.44 + 4.25i)T + (-48.3 - 37.1i)T^{2} \) |
| 67 | \( 1 + (0.225 - 0.197i)T + (8.74 - 66.4i)T^{2} \) |
| 71 | \( 1 + (7.95 + 3.29i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.30 + 1.78i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (2.96 + 11.0i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-9.06 + 4.47i)T + (50.5 - 65.8i)T^{2} \) |
| 89 | \( 1 + (-3.82 - 1.58i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (7.44 + 4.30i)T + (48.5 + 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71937773801434743068510656161, −9.568867573467321547654200196609, −8.984016624215966026748375549557, −8.357313947272101849893946633299, −7.67504771036658417294627033655, −6.70480321912756743238364271511, −5.32566414002281612734438183979, −4.44753600244117892832937685114, −2.62213627666538281491422294690, −1.70967462191941394228042013745,
1.08120245869627621749378957124, 2.34056909640394267240934646624, 3.89518718510575434851191999540, 4.18717318896659267278019763630, 6.55109095505894605942668338218, 7.37459395320750485145547018224, 8.085206950219966886176622654568, 8.555145129472592719075618112756, 9.766679393549454757190986652520, 10.45845431583192290288626202846
