L(s) = 1 | + (1.69 − 0.338i)3-s + (2.15 + 0.592i)5-s + (−2.24 + 1.40i)7-s + (2.77 − 1.15i)9-s + (4.22 − 2.43i)11-s + (−2.53 + 2.53i)13-s + (3.86 + 0.276i)15-s + (−3.56 − 0.955i)17-s + (3.75 + 2.17i)19-s + (−3.33 + 3.14i)21-s + (1.67 − 0.447i)23-s + (4.29 + 2.55i)25-s + (4.31 − 2.89i)27-s − 8.23·29-s + (−4.48 − 7.77i)31-s + ⋯ |
L(s) = 1 | + (0.980 − 0.195i)3-s + (0.964 + 0.265i)5-s + (−0.847 + 0.531i)7-s + (0.923 − 0.383i)9-s + (1.27 − 0.734i)11-s + (−0.702 + 0.702i)13-s + (0.997 + 0.0713i)15-s + (−0.864 − 0.231i)17-s + (0.862 + 0.497i)19-s + (−0.726 + 0.686i)21-s + (0.348 − 0.0933i)23-s + (0.859 + 0.511i)25-s + (0.830 − 0.556i)27-s − 1.52·29-s + (−0.806 − 1.39i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.09426 + 0.0614221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.09426 + 0.0614221i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 + 0.338i)T \) |
| 5 | \( 1 + (-2.15 - 0.592i)T \) |
| 7 | \( 1 + (2.24 - 1.40i)T \) |
good | 11 | \( 1 + (-4.22 + 2.43i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.53 - 2.53i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.56 + 0.955i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 2.17i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.67 + 0.447i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 + (4.48 + 7.77i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.80 - 0.483i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 2.33iT - 41T^{2} \) |
| 43 | \( 1 + (-0.403 + 0.403i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.106 - 0.397i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.150 + 0.562i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.59 - 2.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.05 - 3.55i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.04 - 11.3i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 15.2iT - 71T^{2} \) |
| 73 | \( 1 + (7.18 + 1.92i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.84 - 1.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (11.1 + 11.1i)T + 83iT^{2} \) |
| 89 | \( 1 + (6.60 - 11.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.44 + 1.44i)T + 97iT^{2} \) |
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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
−11.25077316376266470692774030403, −9.875072932644348247257455711198, −9.284292228599078343668338236024, −8.895740475611103321854816951372, −7.35966358908299335235165992661, −6.58740953448283428101663839220, −5.68722932724477110698491932117, −4.01250716659040313489249130359, −2.91739098486022621922271641839, −1.80197849587269848326899480403,
1.65656048726456817098623590627, 3.00615847770833171604584239805, 4.14266238700510480530447036718, 5.32608799648322922101568730829, 6.77762646339557024772131085163, 7.29850942696533750910067911994, 8.803270518763288815091365935609, 9.426874466239633943528640500546, 9.931533479400266888145869211867, 10.92368856082607127057759569009