| L(s) = 1 | + 3·4-s − 8·11-s + 5·16-s + 8·19-s − 4·29-s − 16·31-s + 12·41-s − 24·44-s + 14·49-s − 24·59-s − 4·61-s + 3·64-s + 24·76-s − 32·79-s + 20·89-s − 12·101-s + 4·109-s − 12·116-s + 26·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 2.41·11-s + 5/4·16-s + 1.83·19-s − 0.742·29-s − 2.87·31-s + 1.87·41-s − 3.61·44-s + 2·49-s − 3.12·59-s − 0.512·61-s + 3/8·64-s + 2.75·76-s − 3.60·79-s + 2.11·89-s − 1.19·101-s + 0.383·109-s − 1.11·116-s + 2.36·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.157774718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.157774718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.048680918357962561362169726300, −8.517641943674293797784182963565, −7.82486888709185361752613558423, −7.69560322161885833197188108942, −7.52393746176846508746591489342, −7.12140835295756437528023068807, −7.00447617614658877304628413538, −6.07469959949833306285768906823, −5.83723378317111682167448457465, −5.54280218330479725962546144186, −5.39907101212607343739606137333, −4.68883092438991083895752777445, −4.32228049020070311948922165967, −3.44958934066843342593264545128, −3.31568337448429911585662688768, −2.64181913765770442722963523365, −2.56379785282086248891799181508, −1.84126937801901950504463051252, −1.44684837651882119863734412908, −0.42812667334327442222941437745,
0.42812667334327442222941437745, 1.44684837651882119863734412908, 1.84126937801901950504463051252, 2.56379785282086248891799181508, 2.64181913765770442722963523365, 3.31568337448429911585662688768, 3.44958934066843342593264545128, 4.32228049020070311948922165967, 4.68883092438991083895752777445, 5.39907101212607343739606137333, 5.54280218330479725962546144186, 5.83723378317111682167448457465, 6.07469959949833306285768906823, 7.00447617614658877304628413538, 7.12140835295756437528023068807, 7.52393746176846508746591489342, 7.69560322161885833197188108942, 7.82486888709185361752613558423, 8.517641943674293797784182963565, 9.048680918357962561362169726300
