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
−12.36813288981667689391259475098, −11.91472521473380805810296530000, −10.80611140674210017377536918686, −9.383193699127858669008096856049, −7.897083729462121944567840717136, −7.13377714784082780319879192330, −6.36079050721313834303122141624, −5.55562997989613916941042311369, −4.03748161970917794184061789470, −2.63699585010640133969101265031,
0.63841410899674803796070918369, 3.43877053287479605915823324054, 4.13664350518982689485056227406, 5.30103952350778607307398406533, 6.08175547722633893339064584591, 7.59456673536133046315488788510, 9.191053100253366685215952893117, 10.02185449492557050402418535629, 10.93284848630286149796482346667, 11.57443756969632030639705584502