| L(s) = 1 | − 2·5-s + 9-s − 4·23-s − 25-s + 4·31-s − 12·37-s − 2·45-s + 8·47-s + 2·49-s + 8·53-s + 4·67-s − 8·71-s + 81-s + 8·89-s + 4·97-s − 4·103-s + 8·115-s − 11·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·155-s + 157-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1/3·9-s − 0.834·23-s − 1/5·25-s + 0.718·31-s − 1.97·37-s − 0.298·45-s + 1.16·47-s + 2/7·49-s + 1.09·53-s + 0.488·67-s − 0.949·71-s + 1/9·81-s + 0.847·89-s + 0.406·97-s − 0.394·103-s + 0.746·115-s − 121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.642·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60595733779759845935230014944, −7.16859998658918698890744906448, −6.95492718471119060817068802471, −6.32929951628676676958645791241, −5.91147336858088294684780259803, −5.43972812714454128506495989750, −4.93280720261897174117080335233, −4.45082689700355708053621023596, −3.91092225095545530492559604866, −3.70995705786406482503506662772, −3.06141095527443095958675459331, −2.38055204871445229765910787377, −1.82216671043570550922978075877, −0.956022644515000388846167638599, 0,
0.956022644515000388846167638599, 1.82216671043570550922978075877, 2.38055204871445229765910787377, 3.06141095527443095958675459331, 3.70995705786406482503506662772, 3.91092225095545530492559604866, 4.45082689700355708053621023596, 4.93280720261897174117080335233, 5.43972812714454128506495989750, 5.91147336858088294684780259803, 6.32929951628676676958645791241, 6.95492718471119060817068802471, 7.16859998658918698890744906448, 7.60595733779759845935230014944
