| L(s) = 1 | + 2·5-s − 4·7-s − 9-s + 10·13-s + 12·23-s − 7·25-s − 2·29-s − 8·35-s − 2·45-s − 2·49-s + 10·53-s − 28·59-s + 4·63-s + 20·65-s − 8·67-s − 16·71-s − 8·81-s + 4·83-s − 40·91-s − 16·103-s + 20·107-s + 18·109-s + 24·115-s − 10·117-s + 15·121-s − 26·125-s + 127-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 1/3·9-s + 2.77·13-s + 2.50·23-s − 7/5·25-s − 0.371·29-s − 1.35·35-s − 0.298·45-s − 2/7·49-s + 1.37·53-s − 3.64·59-s + 0.503·63-s + 2.48·65-s − 0.977·67-s − 1.89·71-s − 8/9·81-s + 0.439·83-s − 4.19·91-s − 1.57·103-s + 1.93·107-s + 1.72·109-s + 2.23·115-s − 0.924·117-s + 1.36·121-s − 2.32·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.126402568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126402568\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 151 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
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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
−13.82837841297672472574810458594, −13.34924684784480913552501031541, −12.90690889584280208748276950296, −12.57156156737379614916407131246, −11.46404793409669965356109809442, −11.34773848136740402885803148359, −10.56442536522991048073397422435, −10.25475190170627145434171778233, −9.399564312813529369362579074415, −9.090378683942619914044714318202, −8.742874493889395478290563940354, −7.87529861398798680728963547354, −7.06487408227525129793152854033, −6.35570806238188015824100744160, −6.02071832667706757065760761607, −5.61983243082601145682414998739, −4.43739594559277891528076962187, −3.39739732224506275679364929632, −3.08808685831935640145183700853, −1.50679528450003839441860314788,
1.50679528450003839441860314788, 3.08808685831935640145183700853, 3.39739732224506275679364929632, 4.43739594559277891528076962187, 5.61983243082601145682414998739, 6.02071832667706757065760761607, 6.35570806238188015824100744160, 7.06487408227525129793152854033, 7.87529861398798680728963547354, 8.742874493889395478290563940354, 9.090378683942619914044714318202, 9.399564312813529369362579074415, 10.25475190170627145434171778233, 10.56442536522991048073397422435, 11.34773848136740402885803148359, 11.46404793409669965356109809442, 12.57156156737379614916407131246, 12.90690889584280208748276950296, 13.34924684784480913552501031541, 13.82837841297672472574810458594
