| L(s) = 1 | − 2·5-s + 12·11-s + 8·19-s − 25-s + 4·29-s + 4·31-s − 12·41-s − 49-s − 24·55-s + 8·59-s − 24·61-s + 24·71-s − 32·79-s + 28·89-s − 16·95-s − 12·101-s − 28·109-s + 86·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s − 8·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.61·11-s + 1.83·19-s − 1/5·25-s + 0.742·29-s + 0.718·31-s − 1.87·41-s − 1/7·49-s − 3.23·55-s + 1.04·59-s − 3.07·61-s + 2.84·71-s − 3.60·79-s + 2.96·89-s − 1.64·95-s − 1.19·101-s − 2.68·109-s + 7.81·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.579785172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.579785172\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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
−8.347878784668809701242172101138, −8.155807575656471432744686544768, −7.61092480550294512553826920740, −7.36095340221376516654939139371, −6.83080839185318661308765045518, −6.74441008379582629140608651839, −6.23531266231085847618114554260, −6.20162334697453255935169739523, −5.31528950753101128554555254062, −5.27594188110129571461131574342, −4.46152639399575343392414033538, −4.34371805850461078144904530455, −3.92611637615677545031048206575, −3.58317391838343549071027363000, −3.15858759785414045170008819792, −2.93602546988935205610830885606, −1.88834890380341325451942590923, −1.54687819882890179495122354880, −1.11282886335662339463885988937, −0.59585554502381822404249494613,
0.59585554502381822404249494613, 1.11282886335662339463885988937, 1.54687819882890179495122354880, 1.88834890380341325451942590923, 2.93602546988935205610830885606, 3.15858759785414045170008819792, 3.58317391838343549071027363000, 3.92611637615677545031048206575, 4.34371805850461078144904530455, 4.46152639399575343392414033538, 5.27594188110129571461131574342, 5.31528950753101128554555254062, 6.20162334697453255935169739523, 6.23531266231085847618114554260, 6.74441008379582629140608651839, 6.83080839185318661308765045518, 7.36095340221376516654939139371, 7.61092480550294512553826920740, 8.155807575656471432744686544768, 8.347878784668809701242172101138
