L(s) = 1 | + 2·3-s + 2·5-s + 3·9-s − 4·11-s + 4·15-s + 3·25-s + 4·27-s − 16·31-s − 8·33-s − 4·37-s + 6·45-s + 16·47-s − 14·49-s + 12·53-s − 8·55-s + 24·59-s + 8·67-s + 16·71-s + 6·75-s + 5·81-s + 20·89-s − 32·93-s + 4·97-s − 12·99-s − 8·111-s − 12·113-s + 5·121-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s + 9-s − 1.20·11-s + 1.03·15-s + 3/5·25-s + 0.769·27-s − 2.87·31-s − 1.39·33-s − 0.657·37-s + 0.894·45-s + 2.33·47-s − 2·49-s + 1.64·53-s − 1.07·55-s + 3.12·59-s + 0.977·67-s + 1.89·71-s + 0.692·75-s + 5/9·81-s + 2.11·89-s − 3.31·93-s + 0.406·97-s − 1.20·99-s − 0.759·111-s − 1.12·113-s + 5/11·121-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
\(L(1)\) |
\(\approx\) |
\(3.694297949\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.694297949\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.82509924420319171607901005177, −7.47107963029260271307547882484, −6.89294010622960370374517965040, −6.89270810769313104844169224865, −6.01233132932967091954644036311, −5.59585158632682227968586342616, −5.14697097430576642263450693233, −5.00754822907928208673608959123, −3.95303834850677407956821293662, −3.80035613349284838151879027729, −3.23930613694872092422065115869, −2.46356630345198587679515772045, −2.25002432004047770108316173263, −1.77182976748085837883300948740, −0.72522583805620120659533172797,
0.72522583805620120659533172797, 1.77182976748085837883300948740, 2.25002432004047770108316173263, 2.46356630345198587679515772045, 3.23930613694872092422065115869, 3.80035613349284838151879027729, 3.95303834850677407956821293662, 5.00754822907928208673608959123, 5.14697097430576642263450693233, 5.59585158632682227968586342616, 6.01233132932967091954644036311, 6.89270810769313104844169224865, 6.89294010622960370374517965040, 7.47107963029260271307547882484, 7.82509924420319171607901005177