| L(s) = 1 | + 2.59·2-s − 2.26·3-s + 4.74·4-s − 5-s − 5.87·6-s + 2.73·7-s + 7.11·8-s + 2.11·9-s − 2.59·10-s + 11-s − 10.7·12-s − 5.42·13-s + 7.10·14-s + 2.26·15-s + 8.99·16-s − 5.90·17-s + 5.50·18-s − 7.66·19-s − 4.74·20-s − 6.19·21-s + 2.59·22-s + 0.0405·23-s − 16.0·24-s + 25-s − 14.0·26-s + 1.99·27-s + 12.9·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 1.30·3-s + 2.37·4-s − 0.447·5-s − 2.39·6-s + 1.03·7-s + 2.51·8-s + 0.706·9-s − 0.820·10-s + 0.301·11-s − 3.09·12-s − 1.50·13-s + 1.89·14-s + 0.584·15-s + 2.24·16-s − 1.43·17-s + 1.29·18-s − 1.75·19-s − 1.05·20-s − 1.35·21-s + 0.553·22-s + 0.00844·23-s − 3.28·24-s + 0.200·25-s − 2.76·26-s + 0.383·27-s + 2.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 3 | \( 1 + 2.26T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 13 | \( 1 + 5.42T + 13T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 19 | \( 1 + 7.66T + 19T^{2} \) |
| 23 | \( 1 - 0.0405T + 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 - 8.03T + 37T^{2} \) |
| 41 | \( 1 - 6.03T + 41T^{2} \) |
| 43 | \( 1 + 6.67T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 79 | \( 1 + 7.25T + 79T^{2} \) |
| 83 | \( 1 + 0.670T + 83T^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 + 6.91T + 97T^{2} \) |
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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
−7.65444697266973799929263564269, −6.89978212443176211427307645159, −6.45697651126687182229910915772, −5.57352688516153872758111947590, −5.00904472678722308811279830725, −4.43668491408727135016785388676, −3.97087563283224686423162632652, −2.52411710784538770320326181015, −1.84750836325837646798299034404, 0,
1.84750836325837646798299034404, 2.52411710784538770320326181015, 3.97087563283224686423162632652, 4.43668491408727135016785388676, 5.00904472678722308811279830725, 5.57352688516153872758111947590, 6.45697651126687182229910915772, 6.89978212443176211427307645159, 7.65444697266973799929263564269
