| L(s) = 1 | − 5-s − 3·7-s − 3·11-s − 8·17-s + 12·19-s + 6·23-s + 5·25-s − 2·29-s − 9·31-s + 3·35-s − 4·37-s − 10·41-s − 6·43-s + 6·47-s + 7·49-s − 26·53-s + 3·55-s − 12·59-s − 8·61-s − 6·67-s − 24·71-s + 18·73-s + 9·77-s − 3·83-s + 8·85-s − 28·89-s − 12·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 0.904·11-s − 1.94·17-s + 2.75·19-s + 1.25·23-s + 25-s − 0.371·29-s − 1.61·31-s + 0.507·35-s − 0.657·37-s − 1.56·41-s − 0.914·43-s + 0.875·47-s + 49-s − 3.57·53-s + 0.404·55-s − 1.56·59-s − 1.02·61-s − 0.733·67-s − 2.84·71-s + 2.10·73-s + 1.02·77-s − 0.329·83-s + 0.867·85-s − 2.96·89-s − 1.23·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.685067879348353218316216215978, −8.489153319797829438787568010369, −7.72259152564952967517089289230, −7.52477407668060048299298524339, −7.11693158309662915845982184525, −6.99198308404082401872359759489, −6.32269933801688119986109175117, −6.16890043952399395379181529634, −5.34188048773032411748731885712, −5.27480573604409214460224029556, −4.74753163182569142980445251428, −4.46568371590275876782488462577, −3.60441259534869265298399978246, −3.34578572465081479119223222897, −2.96159984522905296786200894002, −2.70973190262987263803706312392, −1.70696821540476642575069897799, −1.31603265489514064208339017567, 0, 0,
1.31603265489514064208339017567, 1.70696821540476642575069897799, 2.70973190262987263803706312392, 2.96159984522905296786200894002, 3.34578572465081479119223222897, 3.60441259534869265298399978246, 4.46568371590275876782488462577, 4.74753163182569142980445251428, 5.27480573604409214460224029556, 5.34188048773032411748731885712, 6.16890043952399395379181529634, 6.32269933801688119986109175117, 6.99198308404082401872359759489, 7.11693158309662915845982184525, 7.52477407668060048299298524339, 7.72259152564952967517089289230, 8.489153319797829438787568010369, 8.685067879348353218316216215978
