| L(s) = 1 | − 4-s + 2·9-s + 6·11-s + 16-s + 8·19-s + 6·29-s − 2·31-s − 2·36-s − 6·44-s + 13·49-s + 30·59-s − 26·61-s − 64-s − 8·76-s + 8·79-s − 5·81-s − 12·89-s + 12·99-s + 18·101-s − 28·109-s − 6·116-s + 5·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s + 1.80·11-s + 1/4·16-s + 1.83·19-s + 1.11·29-s − 0.359·31-s − 1/3·36-s − 0.904·44-s + 13/7·49-s + 3.90·59-s − 3.32·61-s − 1/8·64-s − 0.917·76-s + 0.900·79-s − 5/9·81-s − 1.27·89-s + 1.20·99-s + 1.79·101-s − 2.68·109-s − 0.557·116-s + 5/11·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.251181159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.251181159\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−10.54541634006610212501995118854, −10.38647607678954729037297434716, −9.789023746212829269236502858853, −9.309024566816859540508558013598, −9.299524572331334031166434606079, −8.639590483965322316017607126919, −8.279021715723857189517346236527, −7.62286265732392386231499567223, −7.14855545326978542343110637644, −6.93543893936234043720817083710, −6.29476915092048769817889466014, −5.82910499375873064003413852779, −5.24324181032387123568365639283, −4.80048871964218044716275128104, −4.03456085125776766304225770892, −3.92209598822427649126366876776, −3.21734294158814495828468795182, −2.47084449387157313813355125967, −1.39959178083319643891731881799, −0.986160448718683558439772957054,
0.986160448718683558439772957054, 1.39959178083319643891731881799, 2.47084449387157313813355125967, 3.21734294158814495828468795182, 3.92209598822427649126366876776, 4.03456085125776766304225770892, 4.80048871964218044716275128104, 5.24324181032387123568365639283, 5.82910499375873064003413852779, 6.29476915092048769817889466014, 6.93543893936234043720817083710, 7.14855545326978542343110637644, 7.62286265732392386231499567223, 8.279021715723857189517346236527, 8.639590483965322316017607126919, 9.299524572331334031166434606079, 9.309024566816859540508558013598, 9.789023746212829269236502858853, 10.38647607678954729037297434716, 10.54541634006610212501995118854
