L(s) = 1 | + (−0.0731 + 0.126i)5-s + (2.33 − 1.25i)7-s + (−0.832 − 1.44i)11-s + (0.0999 + 0.173i)13-s + (−3.13 + 5.43i)17-s + (−3.45 − 5.99i)19-s + (3.09 − 5.35i)23-s + (2.48 + 4.31i)25-s + (2.46 − 4.27i)29-s + 2.51·31-s + (−0.0117 + 0.386i)35-s + (−3.50 − 6.06i)37-s + (−1.15 − 2.00i)41-s + (0.940 − 1.62i)43-s − 1.81·47-s + ⋯ |
L(s) = 1 | + (−0.0327 + 0.0566i)5-s + (0.880 − 0.473i)7-s + (−0.250 − 0.434i)11-s + (0.0277 + 0.0480i)13-s + (−0.760 + 1.31i)17-s + (−0.793 − 1.37i)19-s + (0.644 − 1.11i)23-s + (0.497 + 0.862i)25-s + (0.458 − 0.793i)29-s + 0.452·31-s + (−0.00198 + 0.0653i)35-s + (−0.575 − 0.996i)37-s + (−0.180 − 0.313i)41-s + (0.143 − 0.248i)43-s − 0.264·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0907 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0907 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.599696587\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.599696587\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.33 + 1.25i)T \) |
good | 5 | \( 1 + (0.0731 - 0.126i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.832 + 1.44i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0999 - 0.173i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.13 - 5.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.45 + 5.99i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.46 + 4.27i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.15 + 2.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.940 + 1.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 + (-2.67 + 4.62i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 + 0.678T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + (0.778 - 1.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + (-3.75 + 6.50i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.53 + 7.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.98 - 6.90i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.637841327977670890917458781170, −7.891089997509977553262235256480, −6.96087552020993413451770500010, −6.43678124143901692103605879793, −5.37868433851488537335833218532, −4.58967570615710924991018324094, −3.97110655606360671875348524079, −2.76128202915823273488896191643, −1.83448911239918022148338744024, −0.51704283684771202610820425109,
1.28233338073365597384685054339, 2.28213849382436919798826369214, 3.19927393944202618957259521874, 4.46772964157523049547235688484, 4.92209332145405146868915885996, 5.77980285067461623495364764386, 6.70859767837390894633499685310, 7.43059815319711104037402524087, 8.265355112020781637319732506389, 8.736244934746480829909372994538