| L(s) = 1 | − 3-s + (1 + 2i)5-s + 2i·7-s + 9-s + 6i·11-s + 2·13-s + (−1 − 2i)15-s + 6i·17-s − 4i·19-s − 2i·21-s + 8i·23-s + (−3 + 4i)25-s − 27-s − 8·31-s − 6i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.447 + 0.894i)5-s + 0.755i·7-s + 0.333·9-s + 1.80i·11-s + 0.554·13-s + (−0.258 − 0.516i)15-s + 1.45i·17-s − 0.917i·19-s − 0.436i·21-s + 1.66i·23-s + (−0.600 + 0.800i)25-s − 0.192·27-s − 1.43·31-s − 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437587836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437587836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.130684710735546472698892931101, −7.84953048880639867771713975615, −7.25606210715840810081145232024, −6.58617006404089407323509436272, −5.82704583619405326715213014084, −5.29548469957176748482197818513, −4.25198229439711488115949117055, −3.42067582328660837934116204154, −2.19837898300958541066231917449, −1.66019413597225531302210505812,
0.51242014267888937537038522792, 1.07825183516675888762605413422, 2.49536356185492925617822647012, 3.68435918566779454826221583486, 4.33099300777086362962418548851, 5.38086994955392843423990966407, 5.75500582590622337304831424919, 6.55957079198359568851764637453, 7.40102920453523433446155609914, 8.292700767851739626782211240593
