L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + (0.397 − 2.20i)5-s − 1.00·6-s + (1.66 + 1.66i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−1.27 − 1.83i)10-s − 3.67i·11-s + (−0.707 + 0.707i)12-s + (−4.53 − 4.53i)13-s + 2.35·14-s + (−1.83 + 1.27i)15-s − 1.00·16-s + (4.03 + 4.03i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (0.177 − 0.984i)5-s − 0.408·6-s + (0.630 + 0.630i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.403 − 0.580i)10-s − 1.10i·11-s + (−0.204 + 0.204i)12-s + (−1.25 − 1.25i)13-s + 0.630·14-s + (−0.474 + 0.329i)15-s − 0.250·16-s + (0.978 + 0.978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.498222 - 1.54228i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.498222 - 1.54228i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.397 + 2.20i)T \) |
| 23 | \( 1 + (-1.01 + 4.68i)T \) |
good | 7 | \( 1 + (-1.66 - 1.66i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.67iT - 11T^{2} \) |
| 13 | \( 1 + (4.53 + 4.53i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.03 - 4.03i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.786T + 19T^{2} \) |
| 29 | \( 1 - 4.68iT - 29T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 + (4.59 + 4.59i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 + (-1.71 + 1.71i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.71 - 5.71i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.93 - 5.93i)T - 53iT^{2} \) |
| 59 | \( 1 + 13.2iT - 59T^{2} \) |
| 61 | \( 1 - 3.15iT - 61T^{2} \) |
| 67 | \( 1 + (-3.28 - 3.28i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + (-5.62 - 5.62i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + (-5.05 + 5.05i)T - 83iT^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + (2.18 + 2.18i)T + 97iT^{2} \) |
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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
−10.37662768965052843867120352822, −9.286203572602638081855479903868, −8.356434367926068493176382159775, −7.68697496120193153484615766678, −6.13693559327631268480886141938, −5.39876897110220724238035256220, −4.92334890844150508295427626403, −3.40912912140807420933923872127, −2.09862830658003689294387711958, −0.77172879795868340410790361746,
2.09673319209821113268431501425, 3.50372068063244503302128955980, 4.59000136231497082544348189313, 5.21843319780863899868191672805, 6.48948850753646891323688587760, 7.31970846182210560294350657588, 7.64317060246533608420095480334, 9.486186211189141504122300373140, 9.794142232000115471175405219355, 10.90087598728988714332489559319