L(s) = 1 | + (1.07 − 0.915i)2-s + (−1.47 + 1.47i)3-s + (0.323 − 1.97i)4-s + (−1.55 − 1.61i)5-s + (−0.239 + 2.94i)6-s + (−2.13 − 2.13i)7-s + (−1.45 − 2.42i)8-s − 1.36i·9-s + (−3.14 − 0.317i)10-s − 4.29i·11-s + (2.43 + 3.39i)12-s + (0.707 + 0.707i)13-s + (−4.25 − 0.346i)14-s + (4.67 + 0.0906i)15-s + (−3.79 − 1.27i)16-s + (1.46 − 1.46i)17-s + ⋯ |
L(s) = 1 | + (0.762 − 0.647i)2-s + (−0.853 + 0.853i)3-s + (0.161 − 0.986i)4-s + (−0.693 − 0.720i)5-s + (−0.0979 + 1.20i)6-s + (−0.807 − 0.807i)7-s + (−0.515 − 0.856i)8-s − 0.456i·9-s + (−0.994 − 0.100i)10-s − 1.29i·11-s + (0.704 + 0.980i)12-s + (0.196 + 0.196i)13-s + (−1.13 − 0.0926i)14-s + (1.20 + 0.0234i)15-s + (−0.947 − 0.319i)16-s + (0.355 − 0.355i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.381617 - 0.859865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.381617 - 0.859865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.07 + 0.915i)T \) |
| 5 | \( 1 + (1.55 + 1.61i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.47 - 1.47i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.13 + 2.13i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.29iT - 11T^{2} \) |
| 17 | \( 1 + (-1.46 + 1.46i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 + (4.23 - 4.23i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.88iT - 29T^{2} \) |
| 31 | \( 1 - 2.60iT - 31T^{2} \) |
| 37 | \( 1 + (2.66 - 2.66i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + (-8.10 + 8.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.30 - 4.30i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.75 + 3.75i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.15T + 59T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 + (5.76 + 5.76i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.6iT - 71T^{2} \) |
| 73 | \( 1 + (-7.95 - 7.95i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.366T + 79T^{2} \) |
| 83 | \( 1 + (-6.72 + 6.72i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.0iT - 89T^{2} \) |
| 97 | \( 1 + (4.31 - 4.31i)T - 97iT^{2} \) |
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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
−11.57443756969632030639705584502, −10.93284848630286149796482346667, −10.02185449492557050402418535629, −9.191053100253366685215952893117, −7.59456673536133046315488788510, −6.08175547722633893339064584591, −5.30103952350778607307398406533, −4.13664350518982689485056227406, −3.43877053287479605915823324054, −0.63841410899674803796070918369,
2.63699585010640133969101265031, 4.03748161970917794184061789470, 5.55562997989613916941042311369, 6.36079050721313834303122141624, 7.13377714784082780319879192330, 7.897083729462121944567840717136, 9.383193699127858669008096856049, 10.80611140674210017377536918686, 11.91472521473380805810296530000, 12.36813288981667689391259475098