| L(s) = 1 | + (0.929 + 1.06i)2-s + (0.253 + 0.207i)3-s + (−0.272 + 1.98i)4-s + (1.24 + 4.11i)5-s + (0.0138 + 0.462i)6-s + (0.980 + 0.195i)7-s + (−2.36 + 1.55i)8-s + (−0.564 − 2.83i)9-s + (−3.22 + 5.16i)10-s + (0.0251 − 0.254i)11-s + (−0.480 + 0.444i)12-s + (5.01 + 1.52i)13-s + (0.703 + 1.22i)14-s + (−0.539 + 1.30i)15-s + (−3.85 − 1.07i)16-s + (0.132 + 0.319i)17-s + ⋯ |
| L(s) = 1 | + (0.657 + 0.753i)2-s + (0.146 + 0.119i)3-s + (−0.136 + 0.990i)4-s + (0.558 + 1.84i)5-s + (0.00565 + 0.188i)6-s + (0.370 + 0.0737i)7-s + (−0.836 + 0.548i)8-s + (−0.188 − 0.945i)9-s + (−1.02 + 1.63i)10-s + (0.00757 − 0.0768i)11-s + (−0.138 + 0.128i)12-s + (1.39 + 0.421i)13-s + (0.188 + 0.327i)14-s + (−0.139 + 0.336i)15-s + (−0.962 − 0.269i)16-s + (0.0320 + 0.0774i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.796294 + 2.46518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.796294 + 2.46518i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.929 - 1.06i)T \) |
| 7 | \( 1 + (-0.980 - 0.195i)T \) |
| good | 3 | \( 1 + (-0.253 - 0.207i)T + (0.585 + 2.94i)T^{2} \) |
| 5 | \( 1 + (-1.24 - 4.11i)T + (-4.15 + 2.77i)T^{2} \) |
| 11 | \( 1 + (-0.0251 + 0.254i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-5.01 - 1.52i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-0.132 - 0.319i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.68 - 5.01i)T + (-10.5 + 15.7i)T^{2} \) |
| 23 | \( 1 + (-4.53 + 6.78i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (5.78 - 0.569i)T + (28.4 - 5.65i)T^{2} \) |
| 31 | \( 1 + (7.30 + 7.30i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.20 + 1.17i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (3.03 + 2.02i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-3.43 + 2.81i)T + (8.38 - 42.1i)T^{2} \) |
| 47 | \( 1 + (-3.70 + 1.53i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.24 - 0.516i)T + (51.9 + 10.3i)T^{2} \) |
| 59 | \( 1 + (2.16 - 0.656i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-1.45 + 1.77i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (2.04 - 2.49i)T + (-13.0 - 65.7i)T^{2} \) |
| 71 | \( 1 + (-0.933 + 4.69i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-9.53 + 1.89i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (1.93 + 0.802i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.81 + 0.971i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-5.76 - 8.62i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (11.7 + 11.7i)T + 97iT^{2} \) |
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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
−10.64584878384711093710833098790, −9.465038396934470325879414292867, −8.724446435556585572857962631028, −7.61196082883475328023955949771, −6.86568700461892316077807268257, −6.13269860126050264896129982663, −5.61452435310536730813351816146, −3.84338825128771218141989119652, −3.45938300440209733386879351149, −2.22858546811266402898320440067,
1.09252109990176621975102791367, 1.84044622690592875905978919047, 3.34631655108414948266432784727, 4.54112351714090430725474115381, 5.35419260183443264480975449160, 5.62978375232595116464454763403, 7.25218480703739247250568786808, 8.456310938431583165486448844078, 9.008819224366086571722459939337, 9.697690464940965359889602535343
