| L(s) = 1 | + 3-s − 5-s + 9-s + 2·11-s + 2·13-s − 15-s + 4·19-s + 25-s + 27-s − 4·29-s + 6·31-s + 2·33-s − 8·37-s + 2·39-s − 10·41-s − 10·43-s − 45-s − 6·47-s + 10·53-s − 2·55-s + 4·57-s − 12·59-s − 14·61-s − 2·65-s − 6·67-s + 16·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 1.07·31-s + 0.348·33-s − 1.31·37-s + 0.320·39-s − 1.56·41-s − 1.52·43-s − 0.149·45-s − 0.875·47-s + 1.37·53-s − 0.269·55-s + 0.529·57-s − 1.56·59-s − 1.79·61-s − 0.248·65-s − 0.733·67-s + 1.89·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.53970693391689, −15.25757611604552, −14.71976622953796, −13.97474631287626, −13.63873424594849, −13.23946147762369, −12.30379848633424, −11.99258201430211, −11.46647419092504, −10.80706819351282, −10.15524722716648, −9.667193088346547, −8.968468933262948, −8.548535709322328, −7.972265533062965, −7.376273369663436, −6.729671895061633, −6.263580429728765, −5.300883252792891, −4.801944563228680, −3.944693925627110, −3.427609066609248, −2.927804070368661, −1.793032963431239, −1.255768500814297, 0,
1.255768500814297, 1.793032963431239, 2.927804070368661, 3.427609066609248, 3.944693925627110, 4.801944563228680, 5.300883252792891, 6.263580429728765, 6.729671895061633, 7.376273369663436, 7.972265533062965, 8.548535709322328, 8.968468933262948, 9.667193088346547, 10.15524722716648, 10.80706819351282, 11.46647419092504, 11.99258201430211, 12.30379848633424, 13.23946147762369, 13.63873424594849, 13.97474631287626, 14.71976622953796, 15.25757611604552, 15.53970693391689
