| L(s) = 1 | + i·3-s − 2.82i·5-s − 2.82·7-s − 9-s − 4i·11-s − 5.65i·13-s + 2.82·15-s − 2·17-s + 4i·19-s − 2.82i·21-s + 5.65·23-s − 3.00·25-s − i·27-s − 2.82i·29-s − 8.48·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.26i·5-s − 1.06·7-s − 0.333·9-s − 1.20i·11-s − 1.56i·13-s + 0.730·15-s − 0.485·17-s + 0.917i·19-s − 0.617i·21-s + 1.17·23-s − 0.600·25-s − 0.192i·27-s − 0.525i·29-s − 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.681378 - 0.681378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.681378 - 0.681378i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 2.82iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 11.3iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.92306869392882484147080324064, −10.21624786401804013769402092726, −9.079249102634239362327622980988, −8.694247754189887861826088654208, −7.51160963545188632611326604934, −5.89665677343569916421534970387, −5.42757163728111299870223051784, −4.01576280278593421328541002766, −3.01250857646344909223200469975, −0.63161969456271573036383893493,
2.12688548092163566730867907666, 3.19584874368916845285221376212, 4.59014808961757648683749422747, 6.26220373076718752600619059946, 6.94803412047455459878481713307, 7.31227510997071894143389406489, 9.091707865710564112291081055953, 9.589048740438077970213390282372, 10.86214624597115501074020938941, 11.39651086857171981447994480627
