| L(s) = 1 | − 4-s − 3·9-s − 10·11-s + 16-s − 6·19-s + 12·29-s + 8·31-s + 3·36-s − 22·41-s + 10·44-s + 8·59-s + 4·61-s − 64-s − 20·71-s + 6·76-s + 4·79-s − 22·89-s + 30·99-s + 36·109-s − 12·116-s + 53·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 9-s − 3.01·11-s + 1/4·16-s − 1.37·19-s + 2.22·29-s + 1.43·31-s + 1/2·36-s − 3.43·41-s + 1.50·44-s + 1.04·59-s + 0.512·61-s − 1/8·64-s − 2.37·71-s + 0.688·76-s + 0.450·79-s − 2.33·89-s + 3.01·99-s + 3.44·109-s − 1.11·116-s + 4.81·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.792311736872499710709956111953, −8.480645719412381995950143571902, −8.240062220808379565180363808354, −7.75221844997496956988729162755, −7.22972871033631525050809869205, −6.88813602571735589824037104794, −6.27776705825829606733848795820, −6.07165830844898030545968820545, −5.51659758232396871381930395763, −5.11002405248045217891381302415, −4.83535015245104865470929570050, −4.60487297833844137013602898135, −3.91841885518190209386700625795, −3.15619455694130266348025053715, −2.97191928692346512057964156740, −2.49071683744559344059936404983, −2.12181144244860667873958093578, −1.11675087360072396431518897735, 0, 0,
1.11675087360072396431518897735, 2.12181144244860667873958093578, 2.49071683744559344059936404983, 2.97191928692346512057964156740, 3.15619455694130266348025053715, 3.91841885518190209386700625795, 4.60487297833844137013602898135, 4.83535015245104865470929570050, 5.11002405248045217891381302415, 5.51659758232396871381930395763, 6.07165830844898030545968820545, 6.27776705825829606733848795820, 6.88813602571735589824037104794, 7.22972871033631525050809869205, 7.75221844997496956988729162755, 8.240062220808379565180363808354, 8.480645719412381995950143571902, 8.792311736872499710709956111953
