| L(s) = 1 | − 2·5-s − 2·9-s − 8·13-s + 10·17-s − 7·25-s + 4·29-s − 20·37-s + 12·41-s + 4·45-s − 5·49-s − 16·53-s − 10·61-s + 16·65-s − 30·73-s − 5·81-s − 20·85-s + 32·97-s − 36·101-s + 24·109-s + 4·113-s + 16·117-s − 13·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2/3·9-s − 2.21·13-s + 2.42·17-s − 7/5·25-s + 0.742·29-s − 3.28·37-s + 1.87·41-s + 0.596·45-s − 5/7·49-s − 2.19·53-s − 1.28·61-s + 1.98·65-s − 3.51·73-s − 5/9·81-s − 2.16·85-s + 3.24·97-s − 3.58·101-s + 2.29·109-s + 0.376·113-s + 1.47·117-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−9.829456006504791455536603563268, −9.575094487650744885163867781494, −8.865784190712433894154140882392, −8.148422841499315186159351254036, −7.79236899027639545070642498110, −7.41110136347227768123475878009, −6.97775492717497319205239152750, −5.89152757780490661000300260672, −5.62405161804576690819108289841, −4.85172758929064856672456822692, −4.35808362404597816678481770906, −3.25598629803065603741482539939, −3.08915058635532111035563541614, −1.81226351589066908089801551987, 0,
1.81226351589066908089801551987, 3.08915058635532111035563541614, 3.25598629803065603741482539939, 4.35808362404597816678481770906, 4.85172758929064856672456822692, 5.62405161804576690819108289841, 5.89152757780490661000300260672, 6.97775492717497319205239152750, 7.41110136347227768123475878009, 7.79236899027639545070642498110, 8.148422841499315186159351254036, 8.865784190712433894154140882392, 9.575094487650744885163867781494, 9.829456006504791455536603563268
