L(s) = 1 | + (0.841 − 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (0.959 − 0.281i)5-s + (−0.415 − 0.909i)6-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)10-s + (1.61 + 1.03i)11-s + (−0.841 − 0.540i)12-s + (−0.857 + 0.989i)13-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.118 − 0.258i)17-s + (−0.959 + 0.281i)18-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (0.959 − 0.281i)5-s + (−0.415 − 0.909i)6-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)10-s + (1.61 + 1.03i)11-s + (−0.841 − 0.540i)12-s + (−0.857 + 0.989i)13-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.118 − 0.258i)17-s + (−0.959 + 0.281i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.433381111\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.433381111\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.959 + 0.281i)T \) |
good | 7 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.857 - 0.989i)T + (-0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (-1.61 - 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + 0.284T + T^{2} \) |
| 53 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 61 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 67 | \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
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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
−9.122961480655513836580192865734, −7.76136907701728624731947959975, −7.00320908956273123113282819526, −6.21123883050101284468723484135, −5.98346181783470778349693733237, −4.59366146086786776022890356022, −4.19972142727277361362577216184, −2.68376588863421494832122448232, −2.03061975195430801936243084478, −1.31871175463930720243295647955,
1.96615661009003106580508868886, 3.17826904764794223067736495491, 3.57346483871313754793052890348, 4.64725202362574381712313397067, 5.46463653065576526572682447786, 5.96655886898766081561913549173, 6.68865069617250909622781475484, 7.66192070963438238129581914861, 8.628492624334968905392672004897, 9.119797678062088111492615769381