| L(s) = 1 | − 3-s − 4·4-s − 2·7-s − 2·9-s + 4·12-s − 10·13-s + 12·16-s + 4·19-s + 2·21-s + 5·27-s + 8·28-s − 8·31-s + 8·36-s − 4·37-s + 10·39-s + 20·43-s − 12·48-s + 3·49-s + 40·52-s − 4·57-s + 16·61-s + 4·63-s − 32·64-s + 8·67-s − 4·73-s − 16·76-s − 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 0.755·7-s − 2/3·9-s + 1.15·12-s − 2.77·13-s + 3·16-s + 0.917·19-s + 0.436·21-s + 0.962·27-s + 1.51·28-s − 1.43·31-s + 4/3·36-s − 0.657·37-s + 1.60·39-s + 3.04·43-s − 1.73·48-s + 3/7·49-s + 5.54·52-s − 0.529·57-s + 2.04·61-s + 0.503·63-s − 4·64-s + 0.977·67-s − 0.468·73-s − 1.83·76-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−8.846894389721446221328072634367, −8.214592059880155258547933248387, −7.73929750781885351517852687666, −7.24281290023688107365847358204, −6.93454011637102289378074371468, −5.95865885309513387304829184659, −5.45020178597402395728815947142, −5.36755822629614084385217196812, −4.78686073935120916368506592009, −4.25386150379293419498976865827, −3.67606993779998961598679741611, −2.98704088538274633622207409794, −2.35094220433354059590404276967, −0.75515587387361470267464831173, 0,
0.75515587387361470267464831173, 2.35094220433354059590404276967, 2.98704088538274633622207409794, 3.67606993779998961598679741611, 4.25386150379293419498976865827, 4.78686073935120916368506592009, 5.36755822629614084385217196812, 5.45020178597402395728815947142, 5.95865885309513387304829184659, 6.93454011637102289378074371468, 7.24281290023688107365847358204, 7.73929750781885351517852687666, 8.214592059880155258547933248387, 8.846894389721446221328072634367
