| 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
| \(L(1)\) |
\(\approx\) |
\(2.280407883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.280407883\) |
| \(L(\frac{3}{2})\) |
|
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
−9.135228956756755340730560175918, −8.980965641675229681420485760951, −8.113666811618793737560278178928, −7.82152125199658041092972667976, −7.79503511296817343232972095372, −7.38937934969246520481754328389, −6.81707435813572576012425868100, −6.38930093967998026572851481344, −6.19708117064104016137105881924, −5.37457379923804903991682691797, −5.25441913495893232447339161945, −4.71058831049703925920962516382, −4.39379740500828110400824456630, −3.83612704648026580082800802975, −3.72867528759846932333281601695, −2.80905106029342047193583408829, −2.46538139570973127208713808111, −1.83532172137720674061394215318, −1.19574036531338935622700045764, −0.56792110947444546433223757499,
0.56792110947444546433223757499, 1.19574036531338935622700045764, 1.83532172137720674061394215318, 2.46538139570973127208713808111, 2.80905106029342047193583408829, 3.72867528759846932333281601695, 3.83612704648026580082800802975, 4.39379740500828110400824456630, 4.71058831049703925920962516382, 5.25441913495893232447339161945, 5.37457379923804903991682691797, 6.19708117064104016137105881924, 6.38930093967998026572851481344, 6.81707435813572576012425868100, 7.38937934969246520481754328389, 7.79503511296817343232972095372, 7.82152125199658041092972667976, 8.113666811618793737560278178928, 8.980965641675229681420485760951, 9.135228956756755340730560175918
