| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s − 2·13-s + 16-s − 3·18-s + 4·19-s + 8·23-s − 5·25-s − 2·26-s + 8·29-s − 8·31-s + 32-s − 3·36-s − 8·37-s + 4·38-s − 8·41-s + 4·43-s + 8·46-s + 8·47-s − 5·50-s − 2·52-s + 10·53-s + 8·58-s − 12·59-s − 8·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.554·13-s + 1/4·16-s − 0.707·18-s + 0.917·19-s + 1.66·23-s − 25-s − 0.392·26-s + 1.48·29-s − 1.43·31-s + 0.176·32-s − 1/2·36-s − 1.31·37-s + 0.648·38-s − 1.24·41-s + 0.609·43-s + 1.17·46-s + 1.16·47-s − 0.707·50-s − 0.277·52-s + 1.37·53-s + 1.05·58-s − 1.56·59-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.34612951562099, −14.95022389949213, −14.31737296578856, −13.83730917659387, −13.56782796564482, −12.79443935166205, −12.21567495290406, −11.81722411405080, −11.38311690044992, −10.58206969520784, −10.36779937247235, −9.382565611488252, −8.981235730770135, −8.381963964829852, −7.578503456065338, −7.156664786362052, −6.578629607559805, −5.694863928046418, −5.417098874543795, −4.819897857221045, −4.057635762065836, −3.213486521335796, −2.907797167076729, −2.057944769045559, −1.138957943856753, 0,
1.138957943856753, 2.057944769045559, 2.907797167076729, 3.213486521335796, 4.057635762065836, 4.819897857221045, 5.417098874543795, 5.694863928046418, 6.578629607559805, 7.156664786362052, 7.578503456065338, 8.381963964829852, 8.981235730770135, 9.382565611488252, 10.36779937247235, 10.58206969520784, 11.38311690044992, 11.81722411405080, 12.21567495290406, 12.79443935166205, 13.56782796564482, 13.83730917659387, 14.31737296578856, 14.95022389949213, 15.34612951562099
