| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s − 15-s − 6·17-s + 4·19-s − 8·23-s + 25-s + 27-s + 6·29-s − 8·31-s + 4·33-s + 10·37-s + 6·41-s − 4·43-s − 45-s − 7·49-s − 6·51-s − 10·53-s − 4·55-s + 4·57-s + 4·59-s − 2·61-s − 12·67-s − 8·69-s + 16·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s + 1.64·37-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s − 0.840·51-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s − 1.46·67-s − 0.963·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.01598123153496, −14.32072571677898, −14.17514447085879, −13.60225545352624, −12.88473859593389, −12.52179835717254, −11.85656633977982, −11.32498705479027, −11.07578485542373, −10.15077393032650, −9.658424072134190, −9.178315467226478, −8.727071482399926, −8.019231858809963, −7.674389869120069, −6.955448200869125, −6.377307376734519, −5.989127858635239, −4.945698852387529, −4.406385148893497, −3.895012167322188, −3.342811936566762, −2.521029241253027, −1.852613555160718, −1.077417672743936, 0,
1.077417672743936, 1.852613555160718, 2.521029241253027, 3.342811936566762, 3.895012167322188, 4.406385148893497, 4.945698852387529, 5.989127858635239, 6.377307376734519, 6.955448200869125, 7.674389869120069, 8.019231858809963, 8.727071482399926, 9.178315467226478, 9.658424072134190, 10.15077393032650, 11.07578485542373, 11.32498705479027, 11.85656633977982, 12.52179835717254, 12.88473859593389, 13.60225545352624, 14.17514447085879, 14.32072571677898, 15.01598123153496
