| L(s) = 1 | − 2·3-s + 5-s + 9-s + 11-s − 4·13-s − 2·15-s − 4·19-s − 6·23-s + 25-s + 4·27-s + 6·29-s − 8·31-s − 2·33-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s + 45-s − 6·47-s + 6·53-s + 55-s + 8·57-s − 12·59-s + 2·61-s − 4·65-s + 10·67-s + 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.516·15-s − 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.348·33-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 0.875·47-s + 0.824·53-s + 0.134·55-s + 1.05·57-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 1.22·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.65461859919774, −13.14732512256605, −12.45402671066198, −12.24353660380226, −11.93863879304222, −11.26836863681258, −10.87612256976238, −10.42172311723090, −9.891182591912852, −9.644913959062723, −8.924404309491989, −8.319412400887950, −8.038598424464063, −7.096736321592515, −6.840651073944793, −6.336462283837089, −5.876069998298938, −5.299261009928336, −4.940035468874868, −4.426438209530673, −3.755618504029880, −3.095000760659290, −2.316399852905901, −1.859961272929378, −1.131732157559970, 0, 0,
1.131732157559970, 1.859961272929378, 2.316399852905901, 3.095000760659290, 3.755618504029880, 4.426438209530673, 4.940035468874868, 5.299261009928336, 5.876069998298938, 6.336462283837089, 6.840651073944793, 7.096736321592515, 8.038598424464063, 8.319412400887950, 8.924404309491989, 9.644913959062723, 9.891182591912852, 10.42172311723090, 10.87612256976238, 11.26836863681258, 11.93863879304222, 12.24353660380226, 12.45402671066198, 13.14732512256605, 13.65461859919774
