| L(s) = 1 | − 12·17-s − 8·19-s + 2·25-s − 12·41-s − 8·43-s − 2·49-s − 24·59-s + 8·67-s − 4·73-s + 12·89-s − 4·97-s − 24·107-s + 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.91·17-s − 1.83·19-s + 2/5·25-s − 1.87·41-s − 1.21·43-s − 2/7·49-s − 3.12·59-s + 0.977·67-s − 0.468·73-s + 1.27·89-s − 0.406·97-s − 2.32·107-s + 1.12·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.015531523845900068320152927652, −8.443354404956822832722380798542, −8.057050852209817655105485886173, −7.86852777929663488987876925219, −7.08867804479557123672170829895, −6.68571996650659968785279685501, −6.58327905646539781450949091397, −6.32027506245078211169784598159, −5.73825499223907028216052868382, −5.06182597172327518291954383062, −4.63929477064860348917448634603, −4.63027211397494488448464931756, −3.87610903325282435480477851545, −3.63535021906343364664894789563, −2.77122505365859789286595438287, −2.49260455118983694588862785334, −1.85410503708777745297468497660, −1.49968114299870965218229307264, 0, 0,
1.49968114299870965218229307264, 1.85410503708777745297468497660, 2.49260455118983694588862785334, 2.77122505365859789286595438287, 3.63535021906343364664894789563, 3.87610903325282435480477851545, 4.63027211397494488448464931756, 4.63929477064860348917448634603, 5.06182597172327518291954383062, 5.73825499223907028216052868382, 6.32027506245078211169784598159, 6.58327905646539781450949091397, 6.68571996650659968785279685501, 7.08867804479557123672170829895, 7.86852777929663488987876925219, 8.057050852209817655105485886173, 8.443354404956822832722380798542, 9.015531523845900068320152927652
