L(s) = 1 | + (−0.5 − 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 1.99·6-s + (−2 − 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (−1.5 + 2.59i)11-s + (0.999 + 1.73i)12-s + 5·13-s + (−0.499 + 2.59i)14-s + 1.99·15-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.816·6-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.452 + 0.783i)11-s + (0.288 + 0.499i)12-s + 1.38·13-s + (−0.133 + 0.694i)14-s + 0.516·15-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.699032 - 0.531792i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.699032 - 0.531792i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−14.07200851584952180984754432976, −13.15044261811706142703383224756, −12.69204090063733291776161049702, −10.98886607033032661351288790856, −10.09854726735841160559946814581, −8.669190217470503777638136271838, −7.53085383801062044797924814498, −6.39989130578970503796910265395, −3.75548285742172031241020376086, −1.99133160960014135620033600093,
3.36443215474449269685995637012, 5.15545847896666123364690517603, 6.49009573887889420405119395625, 8.432608378056397329194640698368, 9.115226895573535331806530616824, 10.01262366676259156385878379647, 11.39122308004680081391904868716, 13.16981319807470047694460110414, 13.95269645933841741104857906200, 15.46884703286770629763712996237