| L(s) = 1 | + 2·5-s − 8·17-s − 7·25-s + 4·29-s − 4·37-s + 20·41-s − 5·49-s − 26·53-s + 16·61-s + 18·73-s − 16·85-s − 28·89-s − 18·97-s + 14·101-s − 36·109-s + 20·113-s − 13·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.94·17-s − 7/5·25-s + 0.742·29-s − 0.657·37-s + 3.12·41-s − 5/7·49-s − 3.57·53-s + 2.04·61-s + 2.10·73-s − 1.73·85-s − 2.96·89-s − 1.82·97-s + 1.39·101-s − 3.44·109-s + 1.88·113-s − 1.18·121-s − 2.32·125-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 + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
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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
−8.096235085912272934101439989171, −7.75120135592916568614052087932, −7.00681796643489856555968651557, −6.65325213254225917768927617170, −6.31855619934285825414386333790, −5.81482520780945663550334444725, −5.42474736940027213899015616950, −4.78635677042327797457152601343, −4.30284121931526434248881423255, −3.94091986861202608778869492074, −3.10596268507022717082044228879, −2.42238256260612007081644598630, −2.09771190244944839202554321983, −1.30229225574123782841215729642, 0,
1.30229225574123782841215729642, 2.09771190244944839202554321983, 2.42238256260612007081644598630, 3.10596268507022717082044228879, 3.94091986861202608778869492074, 4.30284121931526434248881423255, 4.78635677042327797457152601343, 5.42474736940027213899015616950, 5.81482520780945663550334444725, 6.31855619934285825414386333790, 6.65325213254225917768927617170, 7.00681796643489856555968651557, 7.75120135592916568614052087932, 8.096235085912272934101439989171
