L(s) = 1 | + (−0.681 + 1.23i)2-s + (−1.07 − 1.68i)4-s − 5-s − 1.41i·7-s + (2.82 − 0.175i)8-s + (0.681 − 1.23i)10-s + 6.37i·11-s + 3.54i·13-s + (1.75 + 0.964i)14-s + (−1.70 + 3.61i)16-s + 3.92i·17-s + 1.27·19-s + (1.07 + 1.68i)20-s + (−7.89 − 4.34i)22-s − 6.28·23-s + ⋯ |
L(s) = 1 | + (−0.482 + 0.876i)2-s + (−0.535 − 0.844i)4-s − 0.447·5-s − 0.534i·7-s + (0.998 − 0.0619i)8-s + (0.215 − 0.391i)10-s + 1.92i·11-s + 0.982i·13-s + (0.468 + 0.257i)14-s + (−0.426 + 0.904i)16-s + 0.952i·17-s + 0.292·19-s + (0.239 + 0.377i)20-s + (−1.68 − 0.925i)22-s − 1.31·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.329911 + 0.688797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.329911 + 0.688797i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.681 - 1.23i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 6.37iT - 11T^{2} \) |
| 13 | \( 1 - 3.54iT - 13T^{2} \) |
| 17 | \( 1 - 3.92iT - 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 - 9.00T + 29T^{2} \) |
| 31 | \( 1 - 3.92iT - 31T^{2} \) |
| 37 | \( 1 - 2.51iT - 37T^{2} \) |
| 41 | \( 1 + 5.27iT - 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 - 0.313iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 7.00T + 67T^{2} \) |
| 71 | \( 1 - 0.990T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 8.18iT - 79T^{2} \) |
| 83 | \( 1 + 5.02iT - 83T^{2} \) |
| 89 | \( 1 - 0.386iT - 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−11.81700300262226463502148004029, −10.41780428263952510980891866987, −9.967858206109973557372824507497, −8.876992194406081945604338182896, −7.88008460684720293862226105947, −7.10136270298335600123279703901, −6.33797742153707091631743163896, −4.78498066038201084240610559041, −4.12012216804413939478467720260, −1.73242681721583557075699740113,
0.63993832267415856731392077254, 2.71890742791305965845705278487, 3.55684621164323180575192779133, 5.02799892708519540520587223526, 6.23702677586873992934111308307, 7.88628733009672804189834668659, 8.301886699522291120389988755122, 9.312008125913516902628318859414, 10.29596489636355438450779024451, 11.22856975933242766144794495472