| L(s) = 1 | + 2.46·2-s − 3-s + 4.09·4-s − 0.776·5-s − 2.46·6-s + 2.69·7-s + 5.15·8-s + 9-s − 1.91·10-s − 11-s − 4.09·12-s + 13-s + 6.64·14-s + 0.776·15-s + 4.55·16-s + 2.62·17-s + 2.46·18-s + 6.69·19-s − 3.17·20-s − 2.69·21-s − 2.46·22-s − 8.02·23-s − 5.15·24-s − 4.39·25-s + 2.46·26-s − 27-s + 11.0·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.04·4-s − 0.347·5-s − 1.00·6-s + 1.01·7-s + 1.82·8-s + 0.333·9-s − 0.605·10-s − 0.301·11-s − 1.18·12-s + 0.277·13-s + 1.77·14-s + 0.200·15-s + 1.13·16-s + 0.635·17-s + 0.581·18-s + 1.53·19-s − 0.709·20-s − 0.587·21-s − 0.526·22-s − 1.67·23-s − 1.05·24-s − 0.879·25-s + 0.484·26-s − 0.192·27-s + 2.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.162546524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.162546524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 + 0.776T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 - 6.69T + 19T^{2} \) |
| 23 | \( 1 + 8.02T + 23T^{2} \) |
| 29 | \( 1 + 8.09T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 + 0.398T + 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 + 2.75T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 - 3.79T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 3.73T + 83T^{2} \) |
| 89 | \( 1 - 2.37T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−11.48155448851321885106810612499, −10.87285091859498207683485660396, −9.610201109575882612088063444423, −7.86647572942243790026980120070, −7.34191823636082089700349327628, −5.83913809868824680606584748797, −5.47352530890130098351063332751, −4.35808684881984360956449430405, −3.51190101980306841362668213634, −1.87387695558354055611315603474,
1.87387695558354055611315603474, 3.51190101980306841362668213634, 4.35808684881984360956449430405, 5.47352530890130098351063332751, 5.83913809868824680606584748797, 7.34191823636082089700349327628, 7.86647572942243790026980120070, 9.610201109575882612088063444423, 10.87285091859498207683485660396, 11.48155448851321885106810612499
