| L(s) = 1 | − 1.28·2-s + 3-s − 0.346·4-s − 1.72·5-s − 1.28·6-s − 7-s + 3.01·8-s + 9-s + 2.21·10-s + 4.44·11-s − 0.346·12-s + 3.95·13-s + 1.28·14-s − 1.72·15-s − 3.18·16-s + 3.67·17-s − 1.28·18-s + 3.16·19-s + 0.597·20-s − 21-s − 5.71·22-s − 7.88·23-s + 3.01·24-s − 2.02·25-s − 5.08·26-s + 27-s + 0.346·28-s + ⋯ |
| L(s) = 1 | − 0.909·2-s + 0.577·3-s − 0.173·4-s − 0.770·5-s − 0.524·6-s − 0.377·7-s + 1.06·8-s + 0.333·9-s + 0.700·10-s + 1.33·11-s − 0.100·12-s + 1.09·13-s + 0.343·14-s − 0.445·15-s − 0.796·16-s + 0.892·17-s − 0.303·18-s + 0.726·19-s + 0.133·20-s − 0.218·21-s − 1.21·22-s − 1.64·23-s + 0.615·24-s − 0.405·25-s − 0.997·26-s + 0.192·27-s + 0.0655·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.28T + 2T^{2} \) |
| 5 | \( 1 + 1.72T + 5T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 - 3.67T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 7.88T + 23T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 + 7.40T + 31T^{2} \) |
| 37 | \( 1 + 7.98T + 37T^{2} \) |
| 41 | \( 1 - 8.58T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 - 6.61T + 59T^{2} \) |
| 61 | \( 1 - 2.09T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 1.27T + 83T^{2} \) |
| 89 | \( 1 - 6.30T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−7.64537267693772291737154591762, −7.18895021854256238237121574269, −6.21740385287517964903247393392, −5.51611912612354680045599313743, −4.29913699225023302315082496116, −3.77243519051817413837139819875, −3.40026755060346255527432353402, −1.85302038939590499259603834437, −1.21046505511922393671726274107, 0,
1.21046505511922393671726274107, 1.85302038939590499259603834437, 3.40026755060346255527432353402, 3.77243519051817413837139819875, 4.29913699225023302315082496116, 5.51611912612354680045599313743, 6.21740385287517964903247393392, 7.18895021854256238237121574269, 7.64537267693772291737154591762
