| L(s) = 1 | + 4·5-s + 5·9-s − 6·11-s + 11·25-s − 10·29-s − 4·31-s + 4·41-s + 20·45-s − 49-s − 24·55-s − 20·59-s + 16·61-s + 16·71-s − 10·79-s + 16·81-s − 30·99-s − 24·101-s + 10·109-s + 5·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 5/3·9-s − 1.80·11-s + 11/5·25-s − 1.85·29-s − 0.718·31-s + 0.624·41-s + 2.98·45-s − 1/7·49-s − 3.23·55-s − 2.60·59-s + 2.04·61-s + 1.89·71-s − 1.12·79-s + 16/9·81-s − 3.01·99-s − 2.38·101-s + 0.957·109-s + 5/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.444054032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.444054032\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.324436535970647935784959557627, −9.145210060327404657014522046302, −8.265151592797047247579718580752, −8.192152263829947548463624324076, −7.57654844002445305010518552799, −7.31021401351418615930984807572, −6.86159234660814441517529223325, −6.61934665433711149719088703750, −5.96422198649062102043075658992, −5.54291837291380504301698890474, −5.43752675546078448855074798468, −5.00594796657550242777241291339, −4.32814274497728760755599752704, −4.19175140845424130484500144937, −3.25072636649214047144377526473, −3.00692436585414587288626000558, −2.11849522457810324846160212487, −2.03873493160167086590209888837, −1.51795013975538364452476919141, −0.62456860391467308714401548794,
0.62456860391467308714401548794, 1.51795013975538364452476919141, 2.03873493160167086590209888837, 2.11849522457810324846160212487, 3.00692436585414587288626000558, 3.25072636649214047144377526473, 4.19175140845424130484500144937, 4.32814274497728760755599752704, 5.00594796657550242777241291339, 5.43752675546078448855074798468, 5.54291837291380504301698890474, 5.96422198649062102043075658992, 6.61934665433711149719088703750, 6.86159234660814441517529223325, 7.31021401351418615930984807572, 7.57654844002445305010518552799, 8.192152263829947548463624324076, 8.265151592797047247579718580752, 9.145210060327404657014522046302, 9.324436535970647935784959557627
