L(s) = 1 | + (0.705 + 1.22i)5-s + (1.17 − 2.02i)7-s + (1.30 − 2.25i)11-s + (−1.26 − 2.18i)13-s − 4.94·17-s − 1.00·19-s + (1.50 + 2.60i)23-s + (1.50 − 2.60i)25-s + (−0.0708 + 0.122i)29-s + (−4.77 − 8.26i)31-s + 3.30·35-s − 9.00·37-s + (−4.33 − 7.50i)41-s + (−3.15 + 5.46i)43-s + (3.24 − 5.62i)47-s + ⋯ |
L(s) = 1 | + (0.315 + 0.546i)5-s + (0.442 − 0.766i)7-s + (0.392 − 0.679i)11-s + (−0.350 − 0.606i)13-s − 1.19·17-s − 0.231·19-s + (0.314 + 0.544i)23-s + (0.300 − 0.521i)25-s + (−0.0131 + 0.0228i)29-s + (−0.856 − 1.48i)31-s + 0.558·35-s − 1.48·37-s + (−0.676 − 1.17i)41-s + (−0.481 + 0.833i)43-s + (0.473 − 0.820i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.114983194\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114983194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.705 - 1.22i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 2.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.30 + 2.25i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.26 + 2.18i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 + (-1.50 - 2.60i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0708 - 0.122i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.77 + 8.26i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.00T + 37T^{2} \) |
| 41 | \( 1 + (4.33 + 7.50i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.15 - 5.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.24 + 5.62i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6.02T + 53T^{2} \) |
| 59 | \( 1 + (-5.64 - 9.77i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.45 + 5.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.154 - 0.268i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.24T + 71T^{2} \) |
| 73 | \( 1 + 6.78T + 73T^{2} \) |
| 79 | \( 1 + (4.99 - 8.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.47 - 6.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + (-7.44 + 12.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.437584676903845015418531192461, −7.39579143398865551733783807643, −6.98355785912614983632600909720, −6.10773922828574981133309667806, −5.36307607184818348988928414218, −4.39194997320227720885829605431, −3.65234293570737440750333989964, −2.67556274932062418005694160058, −1.67822675366312809461411220566, −0.31032443936028423445467270601,
1.55765648753198576954938538796, 2.12617938448890692177753571735, 3.31208712679296805744067848035, 4.54430972879685125332758757709, 4.90500808897295585555967810656, 5.75122446272162807262054220123, 6.80045918735081735565689798171, 7.12583746078956659095000537577, 8.456926291095101421980355342468, 8.795249494508015514397064165065