| L(s) = 1 | − 2.63·2-s + 4.91·4-s − 2.28·5-s − 1.04·7-s − 7.67·8-s + 6.01·10-s − 11-s − 13-s + 2.74·14-s + 10.3·16-s + 5.71·17-s + 3.61·19-s − 11.2·20-s + 2.63·22-s − 0.657·23-s + 0.231·25-s + 2.63·26-s − 5.12·28-s + 6.59·29-s − 4.28·31-s − 11.8·32-s − 15.0·34-s + 2.38·35-s + 1.81·37-s − 9.51·38-s + 17.5·40-s − 7.04·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.45·4-s − 1.02·5-s − 0.394·7-s − 2.71·8-s + 1.90·10-s − 0.301·11-s − 0.277·13-s + 0.733·14-s + 2.58·16-s + 1.38·17-s + 0.829·19-s − 2.51·20-s + 0.560·22-s − 0.137·23-s + 0.0463·25-s + 0.515·26-s − 0.969·28-s + 1.22·29-s − 0.770·31-s − 2.09·32-s − 2.57·34-s + 0.403·35-s + 0.297·37-s − 1.54·38-s + 2.77·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 + 0.657T + 23T^{2} \) |
| 29 | \( 1 - 6.59T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 + 7.04T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 - 2.27T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 5.34T + 59T^{2} \) |
| 61 | \( 1 - 2.44T + 61T^{2} \) |
| 67 | \( 1 + 7.54T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 0.468T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 7.84T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.371974184244272566583148044815, −8.272103865026854278089653775773, −7.88362984989706709509547574151, −7.21689757260082440446747508108, −6.37549165940664904519809676894, −5.21666274496620374381071659717, −3.61711575113758732586307010832, −2.75518467669446045954629766634, −1.26331773997224914533503372542, 0,
1.26331773997224914533503372542, 2.75518467669446045954629766634, 3.61711575113758732586307010832, 5.21666274496620374381071659717, 6.37549165940664904519809676894, 7.21689757260082440446747508108, 7.88362984989706709509547574151, 8.272103865026854278089653775773, 9.371974184244272566583148044815
