L(s) = 1 | − 0.904·2-s − 2.66·3-s − 1.18·4-s + 5-s + 2.41·6-s + 3.83·7-s + 2.87·8-s + 4.11·9-s − 0.904·10-s − 3.18·11-s + 3.15·12-s − 2.30·13-s − 3.46·14-s − 2.66·15-s − 0.241·16-s + 17-s − 3.71·18-s + 5.08·19-s − 1.18·20-s − 10.2·21-s + 2.88·22-s + 5.55·23-s − 7.67·24-s + 25-s + 2.08·26-s − 2.96·27-s − 4.53·28-s + ⋯ |
L(s) = 1 | − 0.639·2-s − 1.53·3-s − 0.590·4-s + 0.447·5-s + 0.984·6-s + 1.44·7-s + 1.01·8-s + 1.37·9-s − 0.286·10-s − 0.961·11-s + 0.909·12-s − 0.639·13-s − 0.927·14-s − 0.688·15-s − 0.0602·16-s + 0.242·17-s − 0.876·18-s + 1.16·19-s − 0.264·20-s − 2.23·21-s + 0.614·22-s + 1.15·23-s − 1.56·24-s + 0.200·25-s + 0.409·26-s − 0.570·27-s − 0.856·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 + 0.904T + 2T^{2} \) |
| 3 | \( 1 + 2.66T + 3T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 0.491T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 + 8.13T + 59T^{2} \) |
| 61 | \( 1 + 3.93T + 61T^{2} \) |
| 67 | \( 1 + 16.2T + 67T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 7.67T + 83T^{2} \) |
| 89 | \( 1 + 9.19T + 89T^{2} \) |
| 97 | \( 1 - 6.38T + 97T^{2} \) |
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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
−7.65466661939619234302919545226, −7.24076996517567636152967127557, −6.18211701084904775842250032255, −5.26471579140359780281328774565, −4.98995401421067193670761001574, −4.67040668728425519337815737878, −3.20553798534681168835746658345, −1.78931621253794340791312333803, −1.09734698563960242767092980478, 0,
1.09734698563960242767092980478, 1.78931621253794340791312333803, 3.20553798534681168835746658345, 4.67040668728425519337815737878, 4.98995401421067193670761001574, 5.26471579140359780281328774565, 6.18211701084904775842250032255, 7.24076996517567636152967127557, 7.65466661939619234302919545226