L(s) = 1 | + (1.15 − 1.15i)3-s + 2i·4-s + (−1.65 − 1.5i)5-s + 0.316i·9-s − 3.31·11-s + (2.31 + 2.31i)12-s + (−3.65 + 0.183i)15-s − 4·16-s + (3 − 3.31i)20-s + (6.15 − 6.15i)23-s + (0.5 + 4.97i)25-s + (3.84 + 3.84i)27-s + 9.94·31-s + (−3.84 + 3.84i)33-s − 0.633·36-s + (−8.47 − 8.47i)37-s + ⋯ |
L(s) = 1 | + (0.668 − 0.668i)3-s + i·4-s + (−0.741 − 0.670i)5-s + 0.105i·9-s − 1.00·11-s + (0.668 + 0.668i)12-s + (−0.944 + 0.0473i)15-s − 16-s + (0.670 − 0.741i)20-s + (1.28 − 1.28i)23-s + (0.100 + 0.994i)25-s + (0.739 + 0.739i)27-s + 1.78·31-s + (−0.668 + 0.668i)33-s − 0.105·36-s + (−1.39 − 1.39i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.904114 - 0.0823837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.904114 - 0.0823837i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.65 + 1.5i)T \) |
| 11 | \( 1 + 3.31T \) |
good | 2 | \( 1 - 2iT^{2} \) |
| 3 | \( 1 + (-1.15 + 1.15i)T - 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (-6.15 + 6.15i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 9.94T + 31T^{2} \) |
| 37 | \( 1 + (8.47 + 8.47i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + (2.68 + 2.68i)T + 47iT^{2} \) |
| 53 | \( 1 + (9.63 - 9.63i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 - 9iT - 89T^{2} \) |
| 97 | \( 1 + (13.4 + 13.4i)T + 97iT^{2} \) |
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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
−15.46383208499094676616911500459, −13.89001808452755247293335082366, −12.90018437961591414739990847332, −12.32524711808401709605989414965, −10.88138989192355342832196302463, −8.813174501455526830475471380197, −8.083045099302244928948101749412, −7.17128803580651590665461554075, −4.69283492377536908539427929709, −2.85991361577896401400185087078,
3.14810190884510550788584453401, 4.90586862298137220451443600131, 6.70998683208428034530949216352, 8.278663877808189586967777173745, 9.682303072155755329859385347862, 10.53863727231625196018524193354, 11.66991069215909497491782236321, 13.47315814693768272499698520822, 14.54757051505721561843252976477, 15.41924105522343092015745758834