| L(s) = 1 | + 3·3-s − 18·5-s + 2·7-s + 30·11-s + 65·13-s − 54·15-s + 111·17-s − 46·19-s + 6·21-s − 6·23-s − 7·25-s − 27·27-s + 105·29-s + 200·31-s + 90·33-s − 36·35-s − 17·37-s + 195·39-s + 231·41-s − 514·43-s + 324·47-s + 343·49-s + 333·51-s + 1.27e3·53-s − 540·55-s − 138·57-s + 600·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.60·5-s + 0.107·7-s + 0.822·11-s + 1.38·13-s − 0.929·15-s + 1.58·17-s − 0.555·19-s + 0.0623·21-s − 0.0543·23-s − 0.0559·25-s − 0.192·27-s + 0.672·29-s + 1.15·31-s + 0.474·33-s − 0.173·35-s − 0.0755·37-s + 0.800·39-s + 0.879·41-s − 1.82·43-s + 1.00·47-s + 49-s + 0.914·51-s + 3.31·53-s − 1.32·55-s − 0.320·57-s + 1.32·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.443545800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.443545800\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 9 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 339 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 111 T + 7408 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 46 T - 4743 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T - 12131 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 105 T - 13364 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 17 T - 50364 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 231 T - 15560 T^{2} - 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 514 T + 184689 T^{2} + 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 639 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 600 T + 154621 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 233 T - 172692 T^{2} + 233 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 926 T + 556713 T^{2} - 926 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 930 T + 506989 T^{2} + 930 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 253 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1324 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 498 T - 456965 T^{2} + 498 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + 19 p T + p^{3} T^{2} ) \) |
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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
−10.21759280088295748589653924631, −10.12228070641966794824395147730, −9.608747201184535224727786924457, −8.791863087322136956645716701519, −8.521507810361313081651397849990, −8.440931726225022840282961220656, −7.82734748335094717092900164805, −7.47619997275104978454817976580, −6.96947746243986636099100937777, −6.51612800706993630936987751885, −5.69286067065807715021971876738, −5.68448828610927526421410735388, −4.62523374874596951295113004997, −4.11845221358513004580139850647, −3.76592902164759883703600280283, −3.49818213348786110324906019496, −2.77018826177645645544112788036, −1.98306304860140211172678548734, −1.01552289057209752215427353112, −0.66056249898241450156180098984,
0.66056249898241450156180098984, 1.01552289057209752215427353112, 1.98306304860140211172678548734, 2.77018826177645645544112788036, 3.49818213348786110324906019496, 3.76592902164759883703600280283, 4.11845221358513004580139850647, 4.62523374874596951295113004997, 5.68448828610927526421410735388, 5.69286067065807715021971876738, 6.51612800706993630936987751885, 6.96947746243986636099100937777, 7.47619997275104978454817976580, 7.82734748335094717092900164805, 8.440931726225022840282961220656, 8.521507810361313081651397849990, 8.791863087322136956645716701519, 9.608747201184535224727786924457, 10.12228070641966794824395147730, 10.21759280088295748589653924631
