| L(s) = 1 | − 2.17·2-s − 1.37·3-s + 2.72·4-s + 2.98·6-s − 7-s − 1.57·8-s − 1.11·9-s − 2.11·11-s − 3.74·12-s − 5.19·13-s + 2.17·14-s − 2.01·16-s − 4.07·17-s + 2.42·18-s + 3.65·19-s + 1.37·21-s + 4.60·22-s + 23-s + 2.16·24-s + 11.3·26-s + 5.64·27-s − 2.72·28-s + 5.27·29-s + 9.16·31-s + 7.54·32-s + 2.90·33-s + 8.86·34-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.792·3-s + 1.36·4-s + 1.21·6-s − 0.377·7-s − 0.558·8-s − 0.372·9-s − 0.638·11-s − 1.08·12-s − 1.44·13-s + 0.581·14-s − 0.504·16-s − 0.989·17-s + 0.572·18-s + 0.837·19-s + 0.299·21-s + 0.981·22-s + 0.208·23-s + 0.442·24-s + 2.21·26-s + 1.08·27-s − 0.515·28-s + 0.978·29-s + 1.64·31-s + 1.33·32-s + 0.506·33-s + 1.52·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 + 1.37T + 3T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 + 4.07T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 + 1.57T + 37T^{2} \) |
| 41 | \( 1 - 3.61T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 4.13T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 1.31T + 83T^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.274919882709069990614637963757, −7.35778999689939476916898360502, −6.85718933307676701016007606245, −6.12092671848064587585898834029, −5.11527906311547128527328353636, −4.56214632976043152055219076478, −2.92337832351609162545917809494, −2.31982757747314492227503694036, −0.873851579524839011107359500637, 0,
0.873851579524839011107359500637, 2.31982757747314492227503694036, 2.92337832351609162545917809494, 4.56214632976043152055219076478, 5.11527906311547128527328353636, 6.12092671848064587585898834029, 6.85718933307676701016007606245, 7.35778999689939476916898360502, 8.274919882709069990614637963757
