L(s) = 1 | + (−1 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (3.93 − 3.08i)5-s − 2.44·6-s + (−1.42 − 1.42i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−7.01 − 0.852i)10-s + 3.13·11-s + (2.44 + 2.44i)12-s + (−8.16 + 8.16i)13-s + 2.84i·14-s + (1.04 − 8.59i)15-s − 4·16-s + (−8.80 − 8.80i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.787 − 0.616i)5-s − 0.408·6-s + (−0.203 − 0.203i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.701 − 0.0852i)10-s + 0.284·11-s + (0.204 + 0.204i)12-s + (−0.628 + 0.628i)13-s + 0.203i·14-s + (0.0695 − 0.573i)15-s − 0.250·16-s + (−0.518 − 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.380408934\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.380408934\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (-3.93 + 3.08i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
good | 7 | \( 1 + (1.42 + 1.42i)T + 49iT^{2} \) |
| 11 | \( 1 - 3.13T + 121T^{2} \) |
| 13 | \( 1 + (8.16 - 8.16i)T - 169iT^{2} \) |
| 17 | \( 1 + (8.80 + 8.80i)T + 289iT^{2} \) |
| 19 | \( 1 + 33.0iT - 361T^{2} \) |
| 29 | \( 1 - 8.67iT - 841T^{2} \) |
| 31 | \( 1 + 47.5T + 961T^{2} \) |
| 37 | \( 1 + (-18.6 - 18.6i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 21.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-37.8 + 37.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (4.90 + 4.90i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-6.02 + 6.02i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 91.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-79.6 - 79.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 123.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-89.3 + 89.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 43.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (72.8 - 72.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 85.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (104. + 104. i)T + 9.40e3iT^{2} \) |
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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
−9.467491383297099633230833025251, −9.307063491929439775626771997254, −8.456687950997212876756443855676, −7.21584038975237767272065018407, −6.66582924864621853469216762801, −5.24323555336108881631092609158, −4.22843176137899461378503945758, −2.76660280229097057413517013064, −1.88434906846765747883875751508, −0.52964387724027835313798538446,
1.71733572419678277086860563824, 2.88922303661857238143718189909, 4.15115676644407558565923949641, 5.59377260147874228443006339909, 6.08865268307343498639957001057, 7.28129288729382779290371157389, 7.993696218870107599302801885218, 9.123200800401250786797491682088, 9.626351069621359028283325563073, 10.44707451487961385999485873524