| L(s) = 1 | + 1.70·3-s + (−0.539 − 2.17i)5-s + 2.63i·7-s − 0.0783·9-s − 5.41i·11-s − 6.34·13-s + (−0.921 − 3.70i)15-s − 3.41i·17-s − 3.26i·19-s + 4.49i·21-s − 1.36i·23-s + (−4.41 + 2.34i)25-s − 5.26·27-s − 2i·29-s − 4.68·31-s + ⋯ |
| L(s) = 1 | + 0.986·3-s + (−0.241 − 0.970i)5-s + 0.994i·7-s − 0.0261·9-s − 1.63i·11-s − 1.75·13-s + (−0.237 − 0.957i)15-s − 0.829i·17-s − 0.748i·19-s + 0.981i·21-s − 0.285i·23-s + (−0.883 + 0.468i)25-s − 1.01·27-s − 0.371i·29-s − 0.840·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.352591363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.352591363\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.539 + 2.17i)T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 - 2.63iT - 7T^{2} \) |
| 11 | \( 1 + 5.41iT - 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 + 3.41iT - 17T^{2} \) |
| 19 | \( 1 + 3.26iT - 19T^{2} \) |
| 23 | \( 1 + 1.36iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 - 4.44T + 43T^{2} \) |
| 47 | \( 1 + 4.78iT - 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 + 3.26iT - 59T^{2} \) |
| 61 | \( 1 + 2.49iT - 61T^{2} \) |
| 67 | \( 1 - 7.86T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 - 13.5iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.52T + 89T^{2} \) |
| 97 | \( 1 - 4.58iT - 97T^{2} \) |
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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
−9.211460762088490026808877735936, −8.665364437768847111870982766317, −8.031080447586486175186612647438, −7.20530313822267613727686314847, −5.77665885739156199210369026857, −5.25607739062761704418742915441, −4.15742727427055257248847047664, −2.90260846034973317998639208661, −2.36669993210522031120577544069, −0.45898161306143610099975672638,
1.95454611415069399336704424213, 2.74417669207143747667385092954, 3.84928507030962824134825197254, 4.49520285817842848799570127262, 5.86781359026046147898049134561, 7.20611676137382239056844687602, 7.35220680178105519110396257887, 8.049646905229552278262828247840, 9.352621650423584344939965201940, 9.890015363650159060499460817932
