| L(s) = 1 | − 2·7-s + 9-s − 4·16-s − 2·19-s + 2·23-s − 2·25-s + 14·29-s − 10·37-s − 2·41-s − 10·43-s + 22·47-s − 11·49-s − 4·53-s − 8·59-s − 6·61-s − 2·63-s + 81-s − 10·83-s + 4·97-s + 18·101-s − 14·107-s − 2·109-s + 8·112-s − 10·121-s + 127-s + 131-s + 4·133-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1/3·9-s − 16-s − 0.458·19-s + 0.417·23-s − 2/5·25-s + 2.59·29-s − 1.64·37-s − 0.312·41-s − 1.52·43-s + 3.20·47-s − 1.57·49-s − 0.549·53-s − 1.04·59-s − 0.768·61-s − 0.251·63-s + 1/9·81-s − 1.09·83-s + 0.406·97-s + 1.79·101-s − 1.35·107-s − 0.191·109-s + 0.755·112-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 197 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.658592471026933423139166364079, −8.059030680128112414352610138170, −7.56784557610723627474046110631, −6.96167228096254884536527666829, −6.55394994326854828585095531879, −6.42898611455966732900257287653, −5.67509494678900790309263768533, −5.02154875571795859759006859938, −4.61126274072516194179655250115, −4.11438954922463630443599997117, −3.35691519292663744071897035823, −2.89487443215997130243515356144, −2.19000061718317224934839216052, −1.30084150718422564604810152255, 0,
1.30084150718422564604810152255, 2.19000061718317224934839216052, 2.89487443215997130243515356144, 3.35691519292663744071897035823, 4.11438954922463630443599997117, 4.61126274072516194179655250115, 5.02154875571795859759006859938, 5.67509494678900790309263768533, 6.42898611455966732900257287653, 6.55394994326854828585095531879, 6.96167228096254884536527666829, 7.56784557610723627474046110631, 8.059030680128112414352610138170, 8.658592471026933423139166364079
