| L(s) = 1 | − 1.70·2-s + 0.917·4-s − 1.50·7-s + 1.84·8-s − 1.68·11-s − 4.04·13-s + 2.57·14-s − 4.99·16-s − 0.146·17-s − 3.59·19-s + 2.88·22-s + 4.77·23-s + 6.91·26-s − 1.38·28-s − 6.52·29-s − 31-s + 4.83·32-s + 0.250·34-s − 5.06·37-s + 6.13·38-s − 0.252·41-s − 0.0635·43-s − 1.55·44-s − 8.15·46-s − 0.392·47-s − 4.72·49-s − 3.71·52-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.458·4-s − 0.569·7-s + 0.653·8-s − 0.509·11-s − 1.12·13-s + 0.688·14-s − 1.24·16-s − 0.0355·17-s − 0.823·19-s + 0.615·22-s + 0.994·23-s + 1.35·26-s − 0.261·28-s − 1.21·29-s − 0.179·31-s + 0.854·32-s + 0.0429·34-s − 0.833·37-s + 0.995·38-s − 0.0394·41-s − 0.00968·43-s − 0.233·44-s − 1.20·46-s − 0.0572·47-s − 0.675·49-s − 0.515·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3434704280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3434704280\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 0.146T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 + 0.252T + 41T^{2} \) |
| 43 | \( 1 + 0.0635T + 43T^{2} \) |
| 47 | \( 1 + 0.392T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 7.08T + 59T^{2} \) |
| 61 | \( 1 + 0.825T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 - 7.58T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−7.959310885924399377357179483261, −7.35834861977300880348592996225, −6.92147286106558346091796910865, −5.98565320478978437544315413422, −5.07595207390607277634702440462, −4.46774106856754790986676822328, −3.41064818097563264938207412995, −2.47935395688768244854985499615, −1.66131610355718474689927222349, −0.35284948852840871971586387280,
0.35284948852840871971586387280, 1.66131610355718474689927222349, 2.47935395688768244854985499615, 3.41064818097563264938207412995, 4.46774106856754790986676822328, 5.07595207390607277634702440462, 5.98565320478978437544315413422, 6.92147286106558346091796910865, 7.35834861977300880348592996225, 7.959310885924399377357179483261
