L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.239 + 0.153i)3-s + (0.841 + 0.540i)4-s + (0.273 − 0.0801i)6-s + (−0.654 − 0.755i)8-s + (−0.381 + 0.835i)9-s + (0.857 − 0.989i)11-s − 0.284·12-s + (0.415 + 0.909i)16-s + (1.84 − 0.540i)17-s + (0.601 − 0.694i)18-s + (−0.544 + 1.19i)19-s + (−1.10 + 0.708i)22-s + (0.273 + 0.0801i)24-s + (−0.142 + 0.989i)25-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.239 + 0.153i)3-s + (0.841 + 0.540i)4-s + (0.273 − 0.0801i)6-s + (−0.654 − 0.755i)8-s + (−0.381 + 0.835i)9-s + (0.857 − 0.989i)11-s − 0.284·12-s + (0.415 + 0.909i)16-s + (1.84 − 0.540i)17-s + (0.601 − 0.694i)18-s + (−0.544 + 1.19i)19-s + (−1.10 + 0.708i)22-s + (0.273 + 0.0801i)24-s + (−0.142 + 0.989i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6112928944\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6112928944\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 89 | \( 1 + (0.142 + 0.989i)T \) |
good | 3 | \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 7 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 17 | \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \) |
| 23 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 31 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.68 - 1.08i)T + (0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 53 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \) |
| 61 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \) |
| 97 | \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
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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
−10.68707893138445768099470639255, −9.771919772373541953084473626522, −9.047216818414899703498621572250, −8.030687421860408147252561940694, −7.55659730135463130334456770665, −6.19043364638265820190222187371, −5.54306893715118055257733638774, −3.89664009241904208880875700200, −2.89516130504653282196291442676, −1.35933979859112496892744866388,
1.17257149839545828214491774049, 2.69010799696148106615355746845, 4.15613149177777031602974917886, 5.60504861476571166872551005976, 6.37233929788050969396029730670, 7.15697077638287343643973383513, 8.025629767184901424233806503527, 9.059551634790033165270304369138, 9.586174503266090128624390060783, 10.46066404462533833305497269800