| L(s) = 1 | − i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 2·11-s − 6i·13-s + (−1 + 2i)15-s + 2i·17-s − 2·21-s − 4i·23-s + (3 + 4i)25-s + i·27-s + 8·31-s + 2i·33-s + (−2 + 4i)35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s − 1.66i·13-s + (−0.258 + 0.516i)15-s + 0.485i·17-s − 0.436·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.43·31-s + 0.348i·33-s + (−0.338 + 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.465077 - 0.752510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.465077 - 0.752510i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 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
−11.97204191946259444215266705660, −10.85038935669694538881413320876, −10.13056002929891195176822211936, −8.456148144676839489131086932562, −7.957753317329978975262304578407, −7.00717100446239630133784520713, −5.61924101980997412333828064393, −4.36151973355833603134652685788, −2.98301548675506377793302167956, −0.72864190327182967399520039641,
2.58342719920451962723501294913, 3.94695391081871525948763714951, 5.02781841842898122069153631094, 6.41014651359295177422336904223, 7.53127668903399757597493950167, 8.637805473386390159780841439932, 9.499311722217538316972029861289, 10.60982430904786714834563426049, 11.65746295943836746628078584697, 11.97373279112430878810242545265
