| L(s) = 1 | − 4-s + 6·9-s − 4·11-s + 16-s − 12·29-s − 14·31-s − 6·36-s − 14·41-s + 4·44-s + 28·59-s − 28·61-s − 64-s − 2·71-s + 22·79-s + 27·81-s − 14·89-s − 24·99-s − 24·109-s + 12·116-s − 10·121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·9-s − 1.20·11-s + 1/4·16-s − 2.22·29-s − 2.51·31-s − 36-s − 2.18·41-s + 0.603·44-s + 3.64·59-s − 3.58·61-s − 1/8·64-s − 0.237·71-s + 2.47·79-s + 3·81-s − 1.48·89-s − 2.41·99-s − 2.29·109-s + 1.11·116-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 + 1/2·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.065362765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065362765\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.257661934582560215150225452043, −8.808440855310059206158207446308, −8.338001673029080307914222631235, −7.80861002601228855763598971074, −7.67635116524528384893985820115, −7.26297944906856649589014795244, −6.84293400137670162114374043394, −6.71119916012312423248984022408, −5.80676207351394884804257925465, −5.59233877970865986270635988968, −5.06314045296799055090274489868, −4.98561580711509492099906548385, −4.16432114562072894887066847854, −4.01527410770927735397685532844, −3.50365501487972567466546099342, −3.13037312073223949806135201076, −2.15042791849737324456428576117, −1.88186992031153101738094215586, −1.39394468250655019307469469488, −0.34775555970871776630706154557,
0.34775555970871776630706154557, 1.39394468250655019307469469488, 1.88186992031153101738094215586, 2.15042791849737324456428576117, 3.13037312073223949806135201076, 3.50365501487972567466546099342, 4.01527410770927735397685532844, 4.16432114562072894887066847854, 4.98561580711509492099906548385, 5.06314045296799055090274489868, 5.59233877970865986270635988968, 5.80676207351394884804257925465, 6.71119916012312423248984022408, 6.84293400137670162114374043394, 7.26297944906856649589014795244, 7.67635116524528384893985820115, 7.80861002601228855763598971074, 8.338001673029080307914222631235, 8.808440855310059206158207446308, 9.257661934582560215150225452043
