| L(s) = 1 | − 3-s + 2·5-s + 3·9-s − 7·11-s − 3·13-s − 2·15-s − 5·17-s + 2·19-s − 2·23-s + 3·25-s − 8·27-s − 3·29-s − 16·31-s + 7·33-s − 4·37-s + 3·39-s − 2·41-s + 6·43-s + 6·45-s + 3·47-s + 5·51-s + 10·53-s − 14·55-s − 2·57-s + 16·59-s − 6·61-s − 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 9-s − 2.11·11-s − 0.832·13-s − 0.516·15-s − 1.21·17-s + 0.458·19-s − 0.417·23-s + 3/5·25-s − 1.53·27-s − 0.557·29-s − 2.87·31-s + 1.21·33-s − 0.657·37-s + 0.480·39-s − 0.312·41-s + 0.914·43-s + 0.894·45-s + 0.437·47-s + 0.700·51-s + 1.37·53-s − 1.88·55-s − 0.264·57-s + 2.08·59-s − 0.768·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 126 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.221345665559606901660482760792, −7.84203182632699118213000383556, −7.38608807688222561539267122752, −7.11751261256207745575014903405, −7.05156195854645885923483017149, −6.49490272768468117134143742294, −5.74576289682242546573333651805, −5.62918632422261289477208296107, −5.36873739608471021665711810954, −5.17887727183785227383142718214, −4.49511260737420677607847196876, −4.17016369837727554170397268357, −3.74053866015848729635518726844, −3.13160720684153040146763666133, −2.49426885531844182262658675487, −2.17012654594016786844304897496, −1.93401038058346366834480938811, −1.17694073576242688207767912336, 0, 0,
1.17694073576242688207767912336, 1.93401038058346366834480938811, 2.17012654594016786844304897496, 2.49426885531844182262658675487, 3.13160720684153040146763666133, 3.74053866015848729635518726844, 4.17016369837727554170397268357, 4.49511260737420677607847196876, 5.17887727183785227383142718214, 5.36873739608471021665711810954, 5.62918632422261289477208296107, 5.74576289682242546573333651805, 6.49490272768468117134143742294, 7.05156195854645885923483017149, 7.11751261256207745575014903405, 7.38608807688222561539267122752, 7.84203182632699118213000383556, 8.221345665559606901660482760792
