| L(s) = 1 | − 0.334·2-s + 1.58·3-s − 1.88·4-s − 0.528·6-s + 7-s + 1.29·8-s − 0.503·9-s − 3.57·11-s − 2.98·12-s − 1.89·13-s − 0.334·14-s + 3.34·16-s − 7.97·17-s + 0.168·18-s − 6.70·19-s + 1.58·21-s + 1.19·22-s + 5.99·23-s + 2.05·24-s + 0.631·26-s − 5.53·27-s − 1.88·28-s + 5.12·29-s + 3.36·31-s − 3.71·32-s − 5.64·33-s + 2.66·34-s + ⋯ |
| L(s) = 1 | − 0.236·2-s + 0.912·3-s − 0.944·4-s − 0.215·6-s + 0.377·7-s + 0.459·8-s − 0.167·9-s − 1.07·11-s − 0.861·12-s − 0.524·13-s − 0.0893·14-s + 0.835·16-s − 1.93·17-s + 0.0396·18-s − 1.53·19-s + 0.344·21-s + 0.254·22-s + 1.24·23-s + 0.419·24-s + 0.123·26-s − 1.06·27-s − 0.356·28-s + 0.952·29-s + 0.603·31-s − 0.657·32-s − 0.983·33-s + 0.457·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 2 | \( 1 + 0.334T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 + 1.89T + 13T^{2} \) |
| 17 | \( 1 + 7.97T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 5.99T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 4.98T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 4.04T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 + 8.89T + 59T^{2} \) |
| 61 | \( 1 + 8.06T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 + 0.916T + 71T^{2} \) |
| 73 | \( 1 - 7.36T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 0.484T + 89T^{2} \) |
| 97 | \( 1 - 7.66T + 97T^{2} \) |
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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
−9.477366589544254981400648539008, −8.691969182338648180915360382564, −8.391503793016151661795198914883, −7.48835026189047104527927701059, −6.33024414005057975553912129781, −4.92712836695146551205732668875, −4.47285807335415596767767212924, −3.07343591798856758483913216211, −2.09962391254062220401032877027, 0,
2.09962391254062220401032877027, 3.07343591798856758483913216211, 4.47285807335415596767767212924, 4.92712836695146551205732668875, 6.33024414005057975553912129781, 7.48835026189047104527927701059, 8.391503793016151661795198914883, 8.691969182338648180915360382564, 9.477366589544254981400648539008
