L(s) = 1 | + 2·5-s + (−2.5 − 0.866i)7-s − 2·11-s + (1.5 + 2.59i)13-s + (4 + 6.92i)17-s + (0.5 − 0.866i)19-s − 8·23-s − 25-s + (2 − 3.46i)29-s + (−1.5 + 2.59i)31-s + (−5 − 1.73i)35-s + (0.5 − 0.866i)37-s + (3 + 5.19i)41-s + (−5.5 + 9.52i)43-s + (3 + 5.19i)47-s + ⋯ |
L(s) = 1 | + 0.894·5-s + (−0.944 − 0.327i)7-s − 0.603·11-s + (0.416 + 0.720i)13-s + (0.970 + 1.68i)17-s + (0.114 − 0.198i)19-s − 1.66·23-s − 0.200·25-s + (0.371 − 0.643i)29-s + (−0.269 + 0.466i)31-s + (−0.845 − 0.292i)35-s + (0.0821 − 0.142i)37-s + (0.468 + 0.811i)41-s + (−0.838 + 1.45i)43-s + (0.437 + 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.297769049\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.297769049\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.515301336661274854884769863754, −8.344517441363426941166211926395, −7.84179929845912335778906022086, −6.62503749218396645330216282857, −6.14493559053094399184237769290, −5.53513632282713089785832547663, −4.25811942113176240075701198402, −3.52510699373237300541551136865, −2.39962699984127349239247339258, −1.38492303028493436244175965900,
0.43432404837003469094928763357, 2.02380813467065873040384379180, 2.91737663209407363230463544992, 3.71632197440089622758523613608, 5.14810240008522480478616670969, 5.66533765327689565967496171376, 6.30549875346230478326269624513, 7.31060512349369400305750995214, 7.996509242237018635867543329055, 9.000212758456728362621690504352