L(s) = 1 | − 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (−3.16 − 3.86i)5-s + 2.44·6-s + 4.38·7-s + 2.82i·8-s − 2.99·9-s + (−5.47 + 4.47i)10-s + 2.89i·11-s − 3.46i·12-s − 1.58i·13-s − 6.20i·14-s + (6.70 − 5.48i)15-s + 4.00·16-s − 3.92·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (−0.633 − 0.773i)5-s + 0.408·6-s + 0.627·7-s + 0.353i·8-s − 0.333·9-s + (−0.547 + 0.447i)10-s + 0.263i·11-s − 0.288i·12-s − 0.121i·13-s − 0.443i·14-s + (0.446 − 0.365i)15-s + 0.250·16-s − 0.230·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1055209925\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1055209925\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (3.16 + 3.86i)T \) |
| 23 | \( 1 + (22.3 - 5.42i)T \) |
good | 7 | \( 1 - 4.38T + 49T^{2} \) |
| 11 | \( 1 - 2.89iT - 121T^{2} \) |
| 13 | \( 1 + 1.58iT - 169T^{2} \) |
| 17 | \( 1 + 3.92T + 289T^{2} \) |
| 19 | \( 1 + 23.3iT - 361T^{2} \) |
| 29 | \( 1 + 8.38T + 841T^{2} \) |
| 31 | \( 1 + 32.5T + 961T^{2} \) |
| 37 | \( 1 - 49.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 67.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 16.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 84.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 1.17T + 2.80e3T^{2} \) |
| 59 | \( 1 + 103.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 12.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 125.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 27.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 25.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 25.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 110.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 14.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 63.9T + 9.40e3T^{2} \) |
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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
−9.669976103862020882997991421897, −9.061542444317576558990608575611, −8.215876769627482669152371618126, −7.40521319175229218097148054795, −5.81558786055170885614371448908, −4.73999064254766288311365684141, −4.29158994559195622766654456955, −3.06168842295335733931265323246, −1.59708898834017380593498629794, −0.03737455742945731725696020381,
1.83041401130607318784365793458, 3.36641027609818969618100186672, 4.37824341261457948451902220637, 5.67311515182112817740901833629, 6.44556318475967330062673128331, 7.39594532826190578131231114594, 7.969877278079249366737996000196, 8.678498156504498074237280290749, 9.937169904341661946830840754195, 10.82303120976103934430819019240