| L(s) = 1 | + 6·3-s + 21·9-s + 4·11-s + 6·17-s − 2·19-s − 10·25-s + 54·27-s + 24·33-s − 24·41-s + 16·43-s − 13·49-s + 36·51-s − 12·57-s − 10·59-s + 14·67-s + 2·73-s − 60·75-s + 108·81-s + 12·83-s − 8·89-s − 12·97-s + 84·99-s + 10·107-s + 4·113-s − 10·121-s − 144·123-s + 127-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 7·9-s + 1.20·11-s + 1.45·17-s − 0.458·19-s − 2·25-s + 10.3·27-s + 4.17·33-s − 3.74·41-s + 2.43·43-s − 1.85·49-s + 5.04·51-s − 1.58·57-s − 1.30·59-s + 1.71·67-s + 0.234·73-s − 6.92·75-s + 12·81-s + 1.31·83-s − 0.847·89-s − 1.21·97-s + 8.44·99-s + 0.966·107-s + 0.376·113-s − 0.909·121-s − 12.9·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.434052680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.434052680\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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
−8.197820813644998743406881037051, −7.943968172182858592403142779428, −7.74236222044461134007095453218, −7.28048593381483137900855724231, −6.60845811420238685670203067170, −6.39357456655427571414069968224, −5.46776906443032883183491997236, −4.82665740673161335065260299640, −4.08931083553499496867238827018, −3.66834687648261549407687934911, −3.63803508541238097399374767207, −2.94857221124987493994611399801, −2.43573806798025905726670590752, −1.66549496948467502043296521631, −1.52291178817025335850558392010,
1.52291178817025335850558392010, 1.66549496948467502043296521631, 2.43573806798025905726670590752, 2.94857221124987493994611399801, 3.63803508541238097399374767207, 3.66834687648261549407687934911, 4.08931083553499496867238827018, 4.82665740673161335065260299640, 5.46776906443032883183491997236, 6.39357456655427571414069968224, 6.60845811420238685670203067170, 7.28048593381483137900855724231, 7.74236222044461134007095453218, 7.943968172182858592403142779428, 8.197820813644998743406881037051
