| L(s) = 1 | + (1.79 + 0.381i)2-s + (0.229 − 2.18i)3-s + (1.24 + 0.554i)4-s + (−0.425 − 0.472i)5-s + (1.24 − 3.83i)6-s + (−1.28 − 2.31i)7-s + (−0.943 − 0.685i)8-s + (−1.79 − 0.381i)9-s + (−0.582 − 1.00i)10-s + (1.5 − 2.59i)12-s + (0.556 + 1.71i)13-s + (−1.42 − 4.63i)14-s + (−1.13 + 0.821i)15-s + (−3.25 − 3.61i)16-s + (−2.77 + 0.589i)17-s + (−3.07 − 1.36i)18-s + ⋯ |
| L(s) = 1 | + (1.26 + 0.269i)2-s + (0.132 − 1.26i)3-s + (0.623 + 0.277i)4-s + (−0.190 − 0.211i)5-s + (0.508 − 1.56i)6-s + (−0.486 − 0.873i)7-s + (−0.333 − 0.242i)8-s + (−0.598 − 0.127i)9-s + (−0.184 − 0.319i)10-s + (0.433 − 0.750i)12-s + (0.154 + 0.475i)13-s + (−0.381 − 1.23i)14-s + (−0.291 + 0.212i)15-s + (−0.814 − 0.904i)16-s + (−0.672 + 0.142i)17-s + (−0.724 − 0.322i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23191 - 2.10123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23191 - 2.10123i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.28 + 2.31i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.79 - 0.381i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.229 + 2.18i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (0.425 + 0.472i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-0.556 - 1.71i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.77 - 0.589i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-5.08 + 2.26i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.43 - 6.13i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.30 + 4.77i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.634 + 6.03i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-6.09 - 4.42i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + (2.58 - 1.15i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.99 + 5.55i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-10.7 - 4.80i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-2.89 - 3.21i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.801 - 1.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.32 + 4.08i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-14.6 - 6.50i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.65 - 0.990i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.85 + 8.77i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.182 - 0.315i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.802 + 2.46i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.837055554847874539107644059186, −8.996250464429547583335283696702, −7.74872566013222836497998795644, −7.10102056224010069626184405517, −6.50170672706180763933245775310, −5.59224517888074551256383235768, −4.42250993656588180400801226557, −3.65180501394519784084211812406, −2.38372547833422189498850094316, −0.77614870631079079977972715050,
2.47488506954238135567235325387, 3.41226129811447016368760649052, 3.99001202480103720285372694937, 5.11615644765788313114066663027, 5.59745799398636466189720022418, 6.65874829531832668631460642262, 8.047211759239858074256097457392, 9.108300907657552590663593751259, 9.596114059060520441071349436655, 10.61559397019689698339050318354
