| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 15-s + 6·17-s − 4·19-s + 25-s − 27-s + 2·29-s − 8·31-s + 4·33-s − 6·37-s + 10·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s − 6·51-s + 6·53-s − 4·55-s + 4·57-s + 12·59-s − 10·61-s + 4·67-s − 12·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.986·37-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s − 1.28·61-s + 0.488·67-s − 1.42·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.28666011834607, −13.66086963179385, −13.03857655877004, −12.83353538628233, −12.31860520728257, −11.77874811199843, −11.16987127048028, −10.61575799237414, −10.38068837077839, −9.802224866645034, −9.353808733302434, −8.591293168704105, −8.154105425435723, −7.525830474720913, −7.146610661835787, −6.386213770919895, −5.924977294324539, −5.367097880922185, −5.042390744631031, −4.367272706128269, −3.552187325503701, −3.073722318204786, −2.225381745614830, −1.714481333377456, −0.8140464787231013, 0,
0.8140464787231013, 1.714481333377456, 2.225381745614830, 3.073722318204786, 3.552187325503701, 4.367272706128269, 5.042390744631031, 5.367097880922185, 5.924977294324539, 6.386213770919895, 7.146610661835787, 7.525830474720913, 8.154105425435723, 8.591293168704105, 9.353808733302434, 9.802224866645034, 10.38068837077839, 10.61575799237414, 11.16987127048028, 11.77874811199843, 12.31860520728257, 12.83353538628233, 13.03857655877004, 13.66086963179385, 14.28666011834607
