| L(s) = 1 | − 3-s − 2·4-s − 5-s − 3·7-s + 9-s − 3·11-s + 2·12-s − 4·13-s + 15-s + 4·16-s + 8·17-s − 6·19-s + 2·20-s + 3·21-s + 3·23-s − 4·25-s − 27-s + 6·28-s − 29-s + 4·31-s + 3·33-s + 3·35-s − 2·36-s + 37-s + 4·39-s − 10·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s + 1.94·17-s − 1.37·19-s + 0.447·20-s + 0.654·21-s + 0.625·23-s − 4/5·25-s − 0.192·27-s + 1.13·28-s − 0.185·29-s + 0.718·31-s + 0.522·33-s + 0.507·35-s − 1/3·36-s + 0.164·37-s + 0.640·39-s − 1.56·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.66579327489625329570299724335, −11.90753988390856144202458034494, −10.19500279746487908916228357990, −9.896550436222926403305189175813, −8.405267645813230706380265272801, −7.30736849693508971465380420611, −5.83486052932247599921983260982, −4.75349830419579613526576050031, −3.31847810114563157193588872502, 0,
3.31847810114563157193588872502, 4.75349830419579613526576050031, 5.83486052932247599921983260982, 7.30736849693508971465380420611, 8.405267645813230706380265272801, 9.896550436222926403305189175813, 10.19500279746487908916228357990, 11.90753988390856144202458034494, 12.66579327489625329570299724335
