L(s) = 1 | + (−0.0498 + 0.697i)3-s + (−0.540 − 0.841i)4-s + (0.989 + 0.142i)5-s + (0.506 + 0.0727i)9-s + (−0.281 + 0.959i)11-s + (0.613 − 0.334i)12-s + (−0.148 + 0.682i)15-s + (−0.415 + 0.909i)16-s + (−0.415 − 0.909i)20-s + (0.281 − 0.959i)23-s + (0.959 + 0.281i)25-s + (−0.224 + 1.03i)27-s + (1.29 − 1.49i)31-s + (−0.654 − 0.244i)33-s + (−0.212 − 0.465i)36-s + (−0.340 + 0.254i)37-s + ⋯ |
L(s) = 1 | + (−0.0498 + 0.697i)3-s + (−0.540 − 0.841i)4-s + (0.989 + 0.142i)5-s + (0.506 + 0.0727i)9-s + (−0.281 + 0.959i)11-s + (0.613 − 0.334i)12-s + (−0.148 + 0.682i)15-s + (−0.415 + 0.909i)16-s + (−0.415 − 0.909i)20-s + (0.281 − 0.959i)23-s + (0.959 + 0.281i)25-s + (−0.224 + 1.03i)27-s + (1.29 − 1.49i)31-s + (−0.654 − 0.244i)33-s + (−0.212 − 0.465i)36-s + (−0.340 + 0.254i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.144990879\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.144990879\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.989 - 0.142i)T \) |
| 11 | \( 1 + (0.281 - 0.959i)T \) |
| 23 | \( 1 + (-0.281 + 0.959i)T \) |
good | 2 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 3 | \( 1 + (0.0498 - 0.697i)T + (-0.989 - 0.142i)T^{2} \) |
| 7 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 13 | \( 1 + (0.755 + 0.654i)T^{2} \) |
| 17 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 19 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 1.49i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (0.340 - 0.254i)T + (0.281 - 0.959i)T^{2} \) |
| 41 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 43 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 47 | \( 1 + (1.13 - 1.13i)T - iT^{2} \) |
| 53 | \( 1 + (-0.682 - 1.83i)T + (-0.755 + 0.654i)T^{2} \) |
| 59 | \( 1 + (-0.983 + 0.449i)T + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 67 | \( 1 + (1.64 + 0.898i)T + (0.540 + 0.841i)T^{2} \) |
| 71 | \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.909 - 0.415i)T^{2} \) |
| 79 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 89 | \( 1 + (1.19 + 1.37i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (-0.718 + 0.959i)T + (-0.281 - 0.959i)T^{2} \) |
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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
−10.07712842326064643474825162814, −9.396603594252125788052618008895, −8.653180389586278200361117359872, −7.38576435778208870998655154272, −6.43690301474206739486134879796, −5.66456433707727210045337512770, −4.72170972591936716340848531099, −4.30526901495030513182259592159, −2.64569099543400950341829735140, −1.49058253885776570308989091637,
1.25540942747564719508914616496, 2.60733184312583914805467573103, 3.59728746478259739339757306815, 4.81020591369594320013462139294, 5.64422787433221871258931427071, 6.65899265127924459157999226238, 7.31978848630416960455445082037, 8.338963811974787227906333736563, 8.816895135446243702412084147168, 9.791844566880851227297057138624