L(s) = 1 | + (0.0475 + 0.998i)2-s + (1.91 − 0.560i)3-s + (−0.995 + 0.0950i)4-s + (−0.654 − 0.755i)5-s + (0.651 + 1.88i)6-s + (−0.419 − 0.216i)7-s + (−0.142 − 0.989i)8-s + (2.49 − 1.60i)9-s + (0.723 − 0.690i)10-s + (−1.84 + 0.739i)12-s + (0.195 − 0.428i)14-s + (−1.67 − 1.07i)15-s + (0.981 − 0.189i)16-s + (1.71 + 2.41i)18-s + (0.723 + 0.690i)20-s + (−0.921 − 0.177i)21-s + ⋯ |
L(s) = 1 | + (0.0475 + 0.998i)2-s + (1.91 − 0.560i)3-s + (−0.995 + 0.0950i)4-s + (−0.654 − 0.755i)5-s + (0.651 + 1.88i)6-s + (−0.419 − 0.216i)7-s + (−0.142 − 0.989i)8-s + (2.49 − 1.60i)9-s + (0.723 − 0.690i)10-s + (−1.84 + 0.739i)12-s + (0.195 − 0.428i)14-s + (−1.67 − 1.07i)15-s + (0.981 − 0.189i)16-s + (1.71 + 2.41i)18-s + (0.723 + 0.690i)20-s + (−0.921 − 0.177i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.633846788\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.633846788\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0475 - 0.998i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 67 | \( 1 + (-0.580 - 0.814i)T \) |
good | 3 | \( 1 + (-1.91 + 0.560i)T + (0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.419 + 0.216i)T + (0.580 + 0.814i)T^{2} \) |
| 11 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 13 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 17 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 19 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 23 | \( 1 + (0.235 - 0.971i)T + (-0.888 - 0.458i)T^{2} \) |
| 29 | \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.911 - 1.28i)T + (-0.327 - 0.945i)T^{2} \) |
| 43 | \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \) |
| 47 | \( 1 + (-1.34 - 1.28i)T + (0.0475 + 0.998i)T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.428 - 1.23i)T + (-0.786 + 0.618i)T^{2} \) |
| 71 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 73 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 79 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 83 | \( 1 + (1.74 - 0.336i)T + (0.928 - 0.371i)T^{2} \) |
| 89 | \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.553268491754624562243167852013, −8.708358197582639679413876444457, −8.143726174330452804189952725543, −7.54916473280453195311419106771, −6.96681049362851127996108767861, −5.85735501391946376857241456860, −4.41698691631260062343784714789, −3.88143032262980671172947850495, −2.94081092295622857594144116113, −1.30865228374160533406635966033,
2.02031653844003743367190210235, 2.78143762045118689275427371786, 3.58970535359850048843922717869, 4.05523753750937459663775156529, 5.17347594222258903793176222830, 6.87530996867433118504014683890, 7.67755049157559416880092866877, 8.567357761567573740249737646446, 8.948127351062154216107805453471, 9.864345993775856350316747401556