| L(s) = 1 | + (−1.40 + 0.115i)2-s + (−2.53 + 0.634i)3-s + (1.97 − 0.325i)4-s + (−0.285 − 0.134i)5-s + (3.49 − 1.18i)6-s + (0.865 + 2.85i)7-s + (−2.74 + 0.687i)8-s + (3.37 − 1.80i)9-s + (0.417 + 0.157i)10-s + (−1.76 + 0.262i)11-s + (−4.79 + 2.07i)12-s + (0.330 − 0.925i)13-s + (−1.54 − 3.91i)14-s + (0.808 + 0.160i)15-s + (3.78 − 1.28i)16-s + (−0.771 + 0.153i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0816i)2-s + (−1.46 + 0.366i)3-s + (0.986 − 0.162i)4-s + (−0.127 − 0.0603i)5-s + (1.42 − 0.484i)6-s + (0.326 + 1.07i)7-s + (−0.970 + 0.242i)8-s + (1.12 − 0.601i)9-s + (0.132 + 0.0497i)10-s + (−0.532 + 0.0790i)11-s + (−1.38 + 0.599i)12-s + (0.0918 − 0.256i)13-s + (−0.413 − 1.04i)14-s + (0.208 + 0.0415i)15-s + (0.946 − 0.321i)16-s + (−0.187 + 0.0372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00585571 - 0.0165838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00585571 - 0.0165838i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.115i)T \) |
| good | 3 | \( 1 + (2.53 - 0.634i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (0.285 + 0.134i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-0.865 - 2.85i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (1.76 - 0.262i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-0.330 + 0.925i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (0.771 - 0.153i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (5.55 + 5.03i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (6.61 - 0.651i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (4.99 + 6.73i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-5.49 - 2.27i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.0749 - 1.52i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-0.0557 + 0.0678i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (1.80 - 7.20i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (3.66 + 2.44i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-1.78 + 2.40i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-1.17 - 3.28i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (5.50 - 9.18i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (2.97 + 1.78i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-1.06 + 1.99i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (0.0290 - 0.0959i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (9.18 + 13.7i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.710 + 14.4i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-1.76 - 0.173i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (5.09 - 12.3i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.69084317760041745956075606161, −10.63215746715701245620201367384, −9.962548771541806083227887881000, −8.761854099737432052553143874021, −7.85629559984932687049652208169, −6.37260676765551877485122539193, −5.82004782752558228710658634086, −4.61044090782094668327477821054, −2.31037924247084500463105530454, −0.02174365236970685297382945784,
1.66620975899507975731071133190, 4.00841231979420494825509279057, 5.61391614074035209386576027955, 6.54594853275250014182811416778, 7.44172568458165350690029893794, 8.312900246083106626204683967035, 9.888303305341044625489032772358, 10.66172155856024663672583816750, 11.16216260858831256015972457212, 12.09570350965232017167755711436
