| L(s) = 1 | + (0.0749 + 0.0784i)2-s + (1.72 + 0.645i)3-s + (0.0892 − 1.98i)4-s + (−1.81 − 1.31i)5-s + (0.0783 + 0.183i)6-s + (−0.608 + 4.49i)7-s + (0.325 − 0.284i)8-s + (0.284 + 0.248i)9-s + (−0.0326 − 0.241i)10-s + (−2.41 + 1.44i)11-s + (1.43 − 3.36i)12-s + (2.26 + 1.35i)13-s + (−0.398 + 0.289i)14-s + (−2.27 − 3.44i)15-s + (−3.91 − 0.352i)16-s + (2.02 − 6.23i)17-s + ⋯ |
| L(s) = 1 | + (0.0530 + 0.0554i)2-s + (0.993 + 0.372i)3-s + (0.0446 − 0.993i)4-s + (−0.811 − 0.589i)5-s + (0.0320 + 0.0748i)6-s + (−0.230 + 1.69i)7-s + (0.115 − 0.100i)8-s + (0.0949 + 0.0829i)9-s + (−0.0103 − 0.0763i)10-s + (−0.729 + 0.435i)11-s + (0.414 − 0.970i)12-s + (0.628 + 0.375i)13-s + (−0.106 + 0.0772i)14-s + (−0.586 − 0.888i)15-s + (−0.978 − 0.0880i)16-s + (0.491 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07125 - 0.0335894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07125 - 0.0335894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-5.65 + 6.24i)T \) |
| good | 2 | \( 1 + (-0.0749 - 0.0784i)T + (-0.0897 + 1.99i)T^{2} \) |
| 3 | \( 1 + (-1.72 - 0.645i)T + (2.25 + 1.97i)T^{2} \) |
| 5 | \( 1 + (1.81 + 1.31i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.608 - 4.49i)T + (-6.74 - 1.86i)T^{2} \) |
| 11 | \( 1 + (2.41 - 1.44i)T + (5.21 - 9.68i)T^{2} \) |
| 13 | \( 1 + (-2.26 - 1.35i)T + (6.16 + 11.4i)T^{2} \) |
| 17 | \( 1 + (-2.02 + 6.23i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.83 - 2.78i)T + (-7.46 - 17.4i)T^{2} \) |
| 23 | \( 1 + (-0.664 - 0.320i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-1.70 + 0.471i)T + (24.8 - 14.8i)T^{2} \) |
| 31 | \( 1 + (-6.55 + 0.590i)T + (30.5 - 5.53i)T^{2} \) |
| 37 | \( 1 + (-1.96 + 0.945i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (0.206 - 0.904i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-9.88 - 1.79i)T + (40.2 + 15.1i)T^{2} \) |
| 47 | \( 1 + (11.8 - 4.45i)T + (35.3 - 30.9i)T^{2} \) |
| 53 | \( 1 + (-0.533 - 11.8i)T + (-52.7 + 4.75i)T^{2} \) |
| 59 | \( 1 + (-1.24 + 2.91i)T + (-40.7 - 42.6i)T^{2} \) |
| 61 | \( 1 + (0.355 + 2.62i)T + (-58.8 + 16.2i)T^{2} \) |
| 67 | \( 1 + (-0.0241 + 0.538i)T + (-66.7 - 6.00i)T^{2} \) |
| 73 | \( 1 + (3.71 + 3.88i)T + (-3.27 + 72.9i)T^{2} \) |
| 79 | \( 1 + (6.98 - 6.10i)T + (10.6 - 78.2i)T^{2} \) |
| 83 | \( 1 + (0.514 - 1.20i)T + (-57.3 - 59.9i)T^{2} \) |
| 89 | \( 1 + (0.0743 + 1.65i)T + (-88.6 + 7.97i)T^{2} \) |
| 97 | \( 1 + (2.13 + 9.33i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−14.84246763975947993545911627265, −13.89022029679937266759893005625, −12.43413591034195294104960657381, −11.48181790442459608600203206264, −9.777813126981946204583430117733, −9.024492556260099158526589545597, −8.037069106719531116510215910594, −6.02002815954071592526595360334, −4.68430862279357920838944841639, −2.67710420850044500902847396305,
3.10942514218036835316856979385, 3.96161263249746480135461040726, 6.87076647287414813479643738008, 7.87372956728214544036820918746, 8.363010354174771669141696689592, 10.43658241165392188974828465173, 11.27235416795074059033449774882, 12.97047475729519515044932482853, 13.41124347534243799628193416792, 14.51198735612935343772022515401
