| L(s) = 1 | − 1.61·2-s + 2.15·3-s + 0.594·4-s − 1.84·5-s − 3.46·6-s + 2.26·8-s + 1.63·9-s + 2.97·10-s + 4.65·11-s + 1.28·12-s + 1.51·13-s − 3.97·15-s − 4.83·16-s − 17-s − 2.64·18-s − 0.0991·19-s − 1.09·20-s − 7.49·22-s + 0.626·23-s + 4.87·24-s − 1.59·25-s − 2.43·26-s − 2.93·27-s − 3.12·29-s + 6.40·30-s + 8.27·31-s + 3.26·32-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 1.24·3-s + 0.297·4-s − 0.825·5-s − 1.41·6-s + 0.800·8-s + 0.546·9-s + 0.939·10-s + 1.40·11-s + 0.369·12-s + 0.419·13-s − 1.02·15-s − 1.20·16-s − 0.242·17-s − 0.622·18-s − 0.0227·19-s − 0.245·20-s − 1.59·22-s + 0.130·23-s + 0.995·24-s − 0.319·25-s − 0.477·26-s − 0.564·27-s − 0.579·29-s + 1.16·30-s + 1.48·31-s + 0.576·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.148872069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148872069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 11 | \( 1 - 4.65T + 11T^{2} \) |
| 13 | \( 1 - 1.51T + 13T^{2} \) |
| 19 | \( 1 + 0.0991T + 19T^{2} \) |
| 23 | \( 1 - 0.626T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 - 8.27T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 - 9.03T + 43T^{2} \) |
| 47 | \( 1 + 0.312T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 8.34T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 - 4.63T + 71T^{2} \) |
| 73 | \( 1 + 0.530T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 1.80T + 83T^{2} \) |
| 89 | \( 1 + 7.29T + 89T^{2} \) |
| 97 | \( 1 + 5.53T + 97T^{2} \) |
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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
−9.671320953057689228820801356346, −9.319568884773617094658105119645, −8.531652708288378219472759488256, −7.966441770609477174960270309395, −7.29751810099408455962837266206, −6.18261770847954239552605412483, −4.34664522294083028814001442911, −3.80918218833229727625909128800, −2.44173692011715931765812112524, −1.03359193503747592672421039970,
1.03359193503747592672421039970, 2.44173692011715931765812112524, 3.80918218833229727625909128800, 4.34664522294083028814001442911, 6.18261770847954239552605412483, 7.29751810099408455962837266206, 7.966441770609477174960270309395, 8.531652708288378219472759488256, 9.319568884773617094658105119645, 9.671320953057689228820801356346
