| L(s) = 1 | − 4-s − 9-s − 4·11-s + 16-s + 8·29-s + 36-s + 12·41-s + 4·44-s − 2·49-s − 24·59-s − 28·61-s − 64-s − 4·71-s − 16·79-s + 81-s + 16·89-s + 4·99-s − 24·101-s − 28·109-s − 8·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 1.48·29-s + 1/6·36-s + 1.87·41-s + 0.603·44-s − 2/7·49-s − 3.12·59-s − 3.58·61-s − 1/8·64-s − 0.474·71-s − 1.80·79-s + 1/9·81-s + 1.69·89-s + 0.402·99-s − 2.38·101-s − 2.68·109-s − 0.742·116-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4660047342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4660047342\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.048310987488586818230960361768, −8.369749158830849622726163695797, −7.889023824608979117311818396722, −7.76860115865797413948480623479, −7.54491682378360765865229958244, −6.94609393656072801726459869821, −6.36392212570722653978851548852, −6.19594758454851536533325490384, −5.80110155840610961592399105441, −5.33060200352736723320980304836, −4.95330297041993171173896061820, −4.49330831004622263226039870989, −4.36831178494353495923880787683, −3.72083375007140684079021589082, −3.09173744627403608460008293702, −2.69721655858785123884089056511, −2.64156845921894809913343354082, −1.57849636336488535884124184564, −1.24595322760943312660261276553, −0.21512056250361237315298819939,
0.21512056250361237315298819939, 1.24595322760943312660261276553, 1.57849636336488535884124184564, 2.64156845921894809913343354082, 2.69721655858785123884089056511, 3.09173744627403608460008293702, 3.72083375007140684079021589082, 4.36831178494353495923880787683, 4.49330831004622263226039870989, 4.95330297041993171173896061820, 5.33060200352736723320980304836, 5.80110155840610961592399105441, 6.19594758454851536533325490384, 6.36392212570722653978851548852, 6.94609393656072801726459869821, 7.54491682378360765865229958244, 7.76860115865797413948480623479, 7.889023824608979117311818396722, 8.369749158830849622726163695797, 9.048310987488586818230960361768
