| L(s) = 1 | − 1.09·2-s − 1.80·3-s − 0.794·4-s + 1.98·6-s + 0.928·7-s + 3.06·8-s + 0.259·9-s − 0.401·11-s + 1.43·12-s + 6.69·13-s − 1.01·14-s − 1.77·16-s + 1.90·17-s − 0.285·18-s − 0.291·19-s − 1.67·21-s + 0.440·22-s − 2.38·23-s − 5.53·24-s − 7.34·26-s + 4.94·27-s − 0.737·28-s − 2.80·29-s − 3.86·31-s − 4.18·32-s + 0.724·33-s − 2.09·34-s + ⋯ |
| L(s) = 1 | − 0.776·2-s − 1.04·3-s − 0.397·4-s + 0.809·6-s + 0.350·7-s + 1.08·8-s + 0.0866·9-s − 0.120·11-s + 0.414·12-s + 1.85·13-s − 0.272·14-s − 0.444·16-s + 0.461·17-s − 0.0672·18-s − 0.0669·19-s − 0.365·21-s + 0.0939·22-s − 0.497·23-s − 1.13·24-s − 1.44·26-s + 0.952·27-s − 0.139·28-s − 0.520·29-s − 0.694·31-s − 0.739·32-s + 0.126·33-s − 0.358·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.09T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 - 0.928T + 7T^{2} \) |
| 11 | \( 1 + 0.401T + 11T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 + 0.291T + 19T^{2} \) |
| 23 | \( 1 + 2.38T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 + 2.33T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 - 9.23T + 61T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.645T + 79T^{2} \) |
| 83 | \( 1 - 8.84T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 9.30T + 97T^{2} \) |
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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
−8.025098463375991864285209762351, −6.97602846560761237584047585230, −6.33543972483867315324981685170, −5.46959710239325044997949108506, −5.12301620885705065926833993026, −4.05108253037050137862992212012, −3.42674614365891951291912456759, −1.83246093325833617176675106789, −1.05521339363022844799584140378, 0,
1.05521339363022844799584140378, 1.83246093325833617176675106789, 3.42674614365891951291912456759, 4.05108253037050137862992212012, 5.12301620885705065926833993026, 5.46959710239325044997949108506, 6.33543972483867315324981685170, 6.97602846560761237584047585230, 8.025098463375991864285209762351
