| L(s) = 1 | + 2.09·2-s − 2.14·3-s + 2.40·4-s − 4.49·6-s + 0.841·8-s + 1.59·9-s − 11-s − 5.14·12-s + 3.54·13-s − 3.03·16-s − 17-s + 3.35·18-s + 3.45·19-s − 2.09·22-s − 4.05·23-s − 1.80·24-s + 7.43·26-s + 3.00·27-s + 6.49·29-s − 9.80·31-s − 8.05·32-s + 2.14·33-s − 2.09·34-s + 3.83·36-s − 1.94·37-s + 7.24·38-s − 7.60·39-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 1.23·3-s + 1.20·4-s − 1.83·6-s + 0.297·8-s + 0.532·9-s − 0.301·11-s − 1.48·12-s + 0.983·13-s − 0.759·16-s − 0.242·17-s + 0.790·18-s + 0.792·19-s − 0.447·22-s − 0.844·23-s − 0.368·24-s + 1.45·26-s + 0.578·27-s + 1.20·29-s − 1.76·31-s − 1.42·32-s + 0.373·33-s − 0.359·34-s + 0.639·36-s − 0.320·37-s + 1.17·38-s − 1.21·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 3 | \( 1 + 2.14T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 19 | \( 1 - 3.45T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 + 9.80T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 + 2.28T + 41T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 + 2.75T + 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.62682265915701822358186962040, −6.79512860466671455586933613101, −6.12335304500975077787825267018, −5.71125151886351435066400527396, −5.06018820732954061180133049830, −4.37451790684587851799599710367, −3.57882442722009369421863344446, −2.74824255979867534110718706810, −1.46597485952199948199749415477, 0,
1.46597485952199948199749415477, 2.74824255979867534110718706810, 3.57882442722009369421863344446, 4.37451790684587851799599710367, 5.06018820732954061180133049830, 5.71125151886351435066400527396, 6.12335304500975077787825267018, 6.79512860466671455586933613101, 7.62682265915701822358186962040
