| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s − 13-s − 15-s + 17-s + 8·19-s + 25-s + 27-s − 6·29-s − 8·31-s + 4·33-s + 2·37-s − 39-s − 10·41-s + 4·43-s − 45-s − 7·49-s + 51-s − 6·53-s − 4·55-s + 8·57-s − 6·61-s + 65-s + 4·67-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 1.83·19-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s − 49-s + 0.140·51-s − 0.824·53-s − 0.539·55-s + 1.05·57-s − 0.768·61-s + 0.124·65-s + 0.488·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.57577273716011, −14.84358632355514, −14.54575340464630, −14.08840385695517, −13.49464498096826, −12.89129116186400, −12.35418130134922, −11.64166434741362, −11.50693932107628, −10.72155716224026, −9.959784000044673, −9.403776669921324, −9.157128469519131, −8.426758497687493, −7.724618120104165, −7.320913419303253, −6.840133184417444, −5.986059663328706, −5.357534016029813, −4.670945573791562, −3.888660487957054, −3.433053572903813, −2.870931958951448, −1.732960949080457, −1.265039227699219, 0,
1.265039227699219, 1.732960949080457, 2.870931958951448, 3.433053572903813, 3.888660487957054, 4.670945573791562, 5.357534016029813, 5.986059663328706, 6.840133184417444, 7.320913419303253, 7.724618120104165, 8.426758497687493, 9.157128469519131, 9.403776669921324, 9.959784000044673, 10.72155716224026, 11.50693932107628, 11.64166434741362, 12.35418130134922, 12.89129116186400, 13.49464498096826, 14.08840385695517, 14.54575340464630, 14.84358632355514, 15.57577273716011
