L(s) = 1 | − i·3-s − i·5-s + 1.44·7-s − 9-s − 0.540·11-s + 4.25·13-s − 15-s + 7.14i·17-s − 0.941·19-s − 1.44i·21-s + (2.91 − 3.80i)23-s − 25-s + i·27-s + 1.26·29-s + 3.15i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 0.547·7-s − 0.333·9-s − 0.162·11-s + 1.18·13-s − 0.258·15-s + 1.73i·17-s − 0.215·19-s − 0.315i·21-s + (0.607 − 0.794i)23-s − 0.200·25-s + 0.192i·27-s + 0.234·29-s + 0.566i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.167857160\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167857160\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 23 | \( 1 + (-2.91 + 3.80i)T \) |
good | 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 + 0.540T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 - 7.14iT - 17T^{2} \) |
| 19 | \( 1 + 0.941T + 19T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 3.15iT - 31T^{2} \) |
| 37 | \( 1 + 1.30iT - 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 - 6.61iT - 47T^{2} \) |
| 53 | \( 1 - 7.54iT - 53T^{2} \) |
| 59 | \( 1 - 11.1iT - 59T^{2} \) |
| 61 | \( 1 + 2.00iT - 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 + 8.07iT - 71T^{2} \) |
| 73 | \( 1 - 1.91T + 73T^{2} \) |
| 79 | \( 1 + 3.99T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−8.118408748424025531666690550848, −7.64018167724224639118380063176, −6.52669851100108099697245989598, −6.13845150402055229620231123657, −5.33361096303703874823121421061, −4.43278872652478236929969024743, −3.76098871383341900621376180316, −2.67845830509998959074600869524, −1.65139694630035142733947691992, −0.982688731578898687587939757574,
0.69442807778164191278251439408, 1.98698648668124735995178515401, 2.98461610579503910409299717453, 3.63788711825347635478031257956, 4.56234961444175527639208448720, 5.21566347570627929929096028356, 5.92781528720606260276266621261, 6.79090322636573731023022989511, 7.45625506043271247673860075366, 8.227823882928784488108894537803