L(s) = 1 | − 3-s + 2·5-s + 9-s + 2·13-s − 2·15-s − 2·17-s + 4·19-s − 25-s − 27-s − 6·29-s − 6·37-s − 2·39-s + 6·41-s − 8·43-s + 2·45-s − 8·47-s + 2·51-s − 6·53-s − 4·57-s − 12·59-s + 10·61-s + 4·65-s − 16·67-s − 8·71-s + 6·73-s + 75-s + 8·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s + 0.298·45-s − 1.16·47-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s + 1.28·61-s + 0.496·65-s − 1.95·67-s − 0.949·71-s + 0.702·73-s + 0.115·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.31564203739328594644090325207, −6.47027529411749487369198931050, −6.04971904052060391144591025387, −5.34970300453341125588827799288, −4.79214251266415149450553393479, −3.82317645056662731564871340463, −3.06537348665754179094256163812, −1.95740145539638854183184857637, −1.35331378281210831851008429091, 0,
1.35331378281210831851008429091, 1.95740145539638854183184857637, 3.06537348665754179094256163812, 3.82317645056662731564871340463, 4.79214251266415149450553393479, 5.34970300453341125588827799288, 6.04971904052060391144591025387, 6.47027529411749487369198931050, 7.31564203739328594644090325207