| L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s − 6·9-s + 8·11-s − 2·14-s + 5·16-s − 12·18-s + 16·22-s + 16·23-s − 6·25-s − 3·28-s − 20·29-s + 6·32-s − 18·36-s + 12·37-s + 8·43-s + 24·44-s + 32·46-s + 49-s − 12·50-s + 12·53-s − 4·56-s − 40·58-s + 6·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s − 2·9-s + 2.41·11-s − 0.534·14-s + 5/4·16-s − 2.82·18-s + 3.41·22-s + 3.33·23-s − 6/5·25-s − 0.566·28-s − 3.71·29-s + 1.06·32-s − 3·36-s + 1.97·37-s + 1.21·43-s + 3.61·44-s + 4.71·46-s + 1/7·49-s − 1.69·50-s + 1.64·53-s − 0.534·56-s − 5.25·58-s + 0.755·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.159308180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.159308180\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.866721919717987183503254472214, −8.859993684429509937089550201483, −7.85978674489636737433292665467, −7.27140118507887344137703587645, −7.08827055361374002488782088399, −6.19217622188912966190136871625, −6.15808758860020713697644216306, −5.54202809749496778607378449269, −5.21535057877927416914375242112, −4.37213247577605881410699996850, −3.67764363944170895713490289621, −3.61340273083380727429539514001, −2.80007795096300664935178443373, −2.18713180950649411226712364496, −1.07575714032888939674416120242,
1.07575714032888939674416120242, 2.18713180950649411226712364496, 2.80007795096300664935178443373, 3.61340273083380727429539514001, 3.67764363944170895713490289621, 4.37213247577605881410699996850, 5.21535057877927416914375242112, 5.54202809749496778607378449269, 6.15808758860020713697644216306, 6.19217622188912966190136871625, 7.08827055361374002488782088399, 7.27140118507887344137703587645, 7.85978674489636737433292665467, 8.859993684429509937089550201483, 8.866721919717987183503254472214
