| L(s) = 1 | − 3-s − 2·4-s − 7-s − 2·9-s + 2·12-s + 4·16-s − 19-s + 21-s − 4·25-s + 5·27-s + 2·28-s − 6·29-s − 31-s + 4·36-s + 4·37-s + 3·47-s − 4·48-s + 49-s + 3·53-s + 57-s − 6·59-s + 2·63-s − 8·64-s + 4·75-s + 2·76-s + 81-s − 3·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s + 0.577·12-s + 16-s − 0.229·19-s + 0.218·21-s − 4/5·25-s + 0.962·27-s + 0.377·28-s − 1.11·29-s − 0.179·31-s + 2/3·36-s + 0.657·37-s + 0.437·47-s − 0.577·48-s + 1/7·49-s + 0.412·53-s + 0.132·57-s − 0.781·59-s + 0.251·63-s − 64-s + 0.461·75-s + 0.229·76-s + 1/9·81-s − 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5287751719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5287751719\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
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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
−8.571317136595086578774667913708, −8.234838037224975547629896432490, −7.65579064808015555760229173067, −7.35674532022489419701895016482, −6.56769538765322429880366416524, −6.19091491846679003266410898010, −5.72447528120598685479450771855, −5.31263799422326525963136284733, −4.91628633003678994750491452626, −4.06519543556360350666243694435, −3.94252184967373043760488884893, −3.11847500492684040169789161120, −2.52961129657399684087912795983, −1.52054958749639074343901757075, −0.41919445413216567484174974853,
0.41919445413216567484174974853, 1.52054958749639074343901757075, 2.52961129657399684087912795983, 3.11847500492684040169789161120, 3.94252184967373043760488884893, 4.06519543556360350666243694435, 4.91628633003678994750491452626, 5.31263799422326525963136284733, 5.72447528120598685479450771855, 6.19091491846679003266410898010, 6.56769538765322429880366416524, 7.35674532022489419701895016482, 7.65579064808015555760229173067, 8.234838037224975547629896432490, 8.571317136595086578774667913708
