L(s) = 1 | − 1.41·2-s − 1.41·3-s + 2.00·4-s + 2.23i·5-s + 2.00·6-s + 3.16i·7-s − 2.82·8-s − 0.999·9-s − 3.16i·10-s − 2.82·12-s − 4.47i·14-s − 3.16i·15-s + 4.00·16-s + 1.41·18-s + 4.47i·20-s − 4.47i·21-s + ⋯ |
L(s) = 1 | − 1.00·2-s − 0.816·3-s + 1.00·4-s + 0.999i·5-s + 0.816·6-s + 1.19i·7-s − 1.00·8-s − 0.333·9-s − 1.00i·10-s − 0.816·12-s − 1.19i·14-s − 0.816i·15-s + 1.00·16-s + 0.333·18-s + 1.00i·20-s − 0.975i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0166295 - 0.224341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0166295 - 0.224341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (0.707 + 4.74i)T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 3.16iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 15.8iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 9.48iT - 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.47509538722166551412216077714, −10.65270948827587735351107890920, −9.850740232306907135565011911764, −8.835909057883146772203891133517, −8.013726893733238622562632511242, −6.78006224307564753279401646659, −6.17251319612075100745372188596, −5.29075531346884972514629865218, −3.20968750243390607845797402593, −2.14246510466951685612222821822,
0.20799206017771098723357147086, 1.53548520747815171502371226809, 3.57949896653122291297284104214, 5.01383843435505054934393107479, 5.95723607844292289017379201568, 7.03441636049696422037444776896, 7.906870493913685137533407884298, 8.814383756400928479461829283180, 9.761953799319623174152710229548, 10.53310362983840754394909654289