L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.959 + 0.281i)4-s + (0.654 + 0.755i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)9-s + (−0.239 − 0.153i)11-s + (0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)18-s + (−0.118 + 0.258i)22-s + (0.959 − 0.281i)23-s + (−0.841 + 0.540i)25-s + (−0.841 − 0.540i)28-s + (−0.698 − 1.53i)29-s + (−0.654 − 0.755i)32-s + ⋯ |
L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.959 + 0.281i)4-s + (0.654 + 0.755i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)9-s + (−0.239 − 0.153i)11-s + (0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (0.142 − 0.989i)18-s + (−0.118 + 0.258i)22-s + (0.959 − 0.281i)23-s + (−0.841 + 0.540i)25-s + (−0.841 − 0.540i)28-s + (−0.698 − 1.53i)29-s + (−0.654 − 0.755i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8826949935\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8826949935\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-0.959 + 0.281i)T \) |
good | 3 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 5 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 11 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 17 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (-0.425 + 1.45i)T + (-0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (0.273 - 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.817 - 0.708i)T + (0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 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
−10.88740930608063170584363554009, −9.746550694635994754621987598711, −9.230945371645260595595232959276, −8.118127695084178911188175769709, −7.51752071160415773903721518619, −5.92088818112540548120614233413, −4.91725283921593521517753884374, −4.06701421015025900571641161458, −2.68099531168247666146408690782, −1.61004919740896207520322029994,
1.41166859007223781262143576848, 3.65497160539527419559723338526, 4.58900708027981681281685581273, 5.40530295035803749666689886059, 6.73967637196956060179019928601, 7.25710303709181885466813062710, 8.094740585933814248966637715608, 9.021501675901416920960187517231, 9.988054022570110434596938628236, 10.55716886661694276710542243042