| L(s) = 1 | − 3-s + 4-s − 2·7-s + 9-s − 12-s − 2·13-s + 16-s + 2·19-s + 2·21-s − 25-s − 27-s − 2·28-s + 36-s − 10·37-s + 2·39-s + 22·43-s − 48-s + 3·49-s − 2·52-s − 2·57-s + 2·61-s − 2·63-s + 64-s − 12·67-s + 10·73-s + 75-s + 2·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.458·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s − 0.377·28-s + 1/6·36-s − 1.64·37-s + 0.320·39-s + 3.35·43-s − 0.144·48-s + 3/7·49-s − 0.277·52-s − 0.264·57-s + 0.256·61-s − 0.251·63-s + 1/8·64-s − 1.46·67-s + 1.17·73-s + 0.115·75-s + 0.229·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338622169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338622169\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.925758442786961590691428846639, −7.71886793500763207100130548275, −7.20481276136351597364288703843, −6.84627164270088746798526670414, −6.49587991264233814474632939114, −5.86366672737057092676215077124, −5.55921197766306254502353102551, −5.23206193743426850928425903169, −4.25945441857471440765324112942, −4.23861623857583401252172064502, −3.33711352613587716311643994647, −2.89830493976033687883443576173, −2.28508171651994554248500999764, −1.52673365627799806411662380976, −0.56722170613835914134961311701,
0.56722170613835914134961311701, 1.52673365627799806411662380976, 2.28508171651994554248500999764, 2.89830493976033687883443576173, 3.33711352613587716311643994647, 4.23861623857583401252172064502, 4.25945441857471440765324112942, 5.23206193743426850928425903169, 5.55921197766306254502353102551, 5.86366672737057092676215077124, 6.49587991264233814474632939114, 6.84627164270088746798526670414, 7.20481276136351597364288703843, 7.71886793500763207100130548275, 7.925758442786961590691428846639
