| L(s) = 1 | − 4-s − 9-s + 4·11-s + 16-s + 10·29-s + 14·31-s + 36-s + 14·41-s − 4·44-s − 49-s + 10·59-s − 6·61-s − 64-s + 24·71-s − 20·79-s + 81-s + 20·89-s − 4·99-s + 24·101-s − 10·116-s − 10·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1.20·11-s + 1/4·16-s + 1.85·29-s + 2.51·31-s + 1/6·36-s + 2.18·41-s − 0.603·44-s − 1/7·49-s + 1.30·59-s − 0.768·61-s − 1/8·64-s + 2.84·71-s − 2.25·79-s + 1/9·81-s + 2.11·89-s − 0.402·99-s + 2.38·101-s − 0.928·116-s − 0.909·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.221509271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221509271\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.04203206286010946799635660096, −9.808632460466585897765817262420, −9.073971963379388790920887533642, −9.055382572194412748832084444824, −8.420874344238838667094674180146, −8.247290282532576061032369182949, −7.61818271994208060256571797231, −7.31652005858638648870522305525, −6.45844813884723478152021312953, −6.30124024137353588011981718323, −6.19468258290892207301574983040, −5.17112473360551779962615920756, −4.98767384158001621680400013200, −4.36236858430158973689928417783, −4.01042146023349905158645309673, −3.48072915361499806176663674517, −2.68583287761610069750233944860, −2.43335352298461453416479505235, −1.22181019720125602154258706613, −0.811903145087288829486097771831,
0.811903145087288829486097771831, 1.22181019720125602154258706613, 2.43335352298461453416479505235, 2.68583287761610069750233944860, 3.48072915361499806176663674517, 4.01042146023349905158645309673, 4.36236858430158973689928417783, 4.98767384158001621680400013200, 5.17112473360551779962615920756, 6.19468258290892207301574983040, 6.30124024137353588011981718323, 6.45844813884723478152021312953, 7.31652005858638648870522305525, 7.61818271994208060256571797231, 8.247290282532576061032369182949, 8.420874344238838667094674180146, 9.055382572194412748832084444824, 9.073971963379388790920887533642, 9.808632460466585897765817262420, 10.04203206286010946799635660096
