| L(s) = 1 | − 1.73·2-s + 3-s + 0.999·4-s − 1.73·6-s + 1.73·7-s + 1.73·8-s − 2·9-s + 0.999·12-s − 3.46·13-s − 2.99·14-s − 5·16-s + 6.92·17-s + 3.46·18-s − 3.46·19-s + 1.73·21-s + 1.73·24-s + 5.99·26-s − 5·27-s + 1.73·28-s − 8·31-s + 5.19·32-s − 11.9·34-s − 1.99·36-s + 8·37-s + 5.99·38-s − 3.46·39-s + 12.1·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.707·6-s + 0.654·7-s + 0.612·8-s − 0.666·9-s + 0.288·12-s − 0.960·13-s − 0.801·14-s − 1.25·16-s + 1.68·17-s + 0.816·18-s − 0.794·19-s + 0.377·21-s + 0.353·24-s + 1.17·26-s − 0.962·27-s + 0.327·28-s − 1.43·31-s + 0.918·32-s − 2.05·34-s − 0.333·36-s + 1.31·37-s + 0.973·38-s − 0.554·39-s + 1.89·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.327171607554445491106677120366, −7.77380019299085524138115369225, −7.42085387940016343638920601121, −6.19345110913961570096630669202, −5.25911822446654566921719686456, −4.45329178128836373699649684300, −3.31542274072744033238996171940, −2.31968796637188150022728979119, −1.40362957628394311272625414820, 0,
1.40362957628394311272625414820, 2.31968796637188150022728979119, 3.31542274072744033238996171940, 4.45329178128836373699649684300, 5.25911822446654566921719686456, 6.19345110913961570096630669202, 7.42085387940016343638920601121, 7.77380019299085524138115369225, 8.327171607554445491106677120366
