| L(s) = 1 | − 4·5-s + 4·13-s + 4·17-s + 2·25-s + 12·29-s + 20·37-s + 12·41-s + 2·49-s + 12·53-s + 4·61-s − 16·65-s + 28·73-s − 16·85-s + 4·89-s − 4·97-s + 12·101-s − 12·109-s − 4·113-s − 18·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.10·13-s + 0.970·17-s + 2/5·25-s + 2.22·29-s + 3.28·37-s + 1.87·41-s + 2/7·49-s + 1.64·53-s + 0.512·61-s − 1.98·65-s + 3.27·73-s − 1.73·85-s + 0.423·89-s − 0.406·97-s + 1.19·101-s − 1.14·109-s − 0.376·113-s − 1.63·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.848264670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.848264670\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.076512848121596140241215200165, −7.63242500478736670765641851926, −7.36082202049510326881285758548, −6.52256885432452015704757780960, −6.38329769807791755045956385135, −5.82368326970125389675476631831, −5.34050418818343429420051907291, −4.62995407509459210959268189869, −4.28451177805084716850337467950, −3.73824894896581631226090928700, −3.67212844043023358887177272155, −2.71947017805277709590872673137, −2.44235807183272353930685105042, −1.02724387350145696940295704376, −0.795101455562395073664105794433,
0.795101455562395073664105794433, 1.02724387350145696940295704376, 2.44235807183272353930685105042, 2.71947017805277709590872673137, 3.67212844043023358887177272155, 3.73824894896581631226090928700, 4.28451177805084716850337467950, 4.62995407509459210959268189869, 5.34050418818343429420051907291, 5.82368326970125389675476631831, 6.38329769807791755045956385135, 6.52256885432452015704757780960, 7.36082202049510326881285758548, 7.63242500478736670765641851926, 8.076512848121596140241215200165
