| L(s) = 1 | + 3-s + 5·7-s + 2·11-s − 2·13-s + 2·17-s − 5·19-s + 5·21-s + 6·23-s + 5·25-s − 27-s + 16·29-s − 3·31-s + 2·33-s − 9·37-s − 2·39-s + 4·41-s + 2·43-s + 8·47-s + 18·49-s + 2·51-s + 6·53-s − 5·57-s − 6·59-s − 2·61-s + 5·67-s + 6·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s + 0.603·11-s − 0.554·13-s + 0.485·17-s − 1.14·19-s + 1.09·21-s + 1.25·23-s + 25-s − 0.192·27-s + 2.97·29-s − 0.538·31-s + 0.348·33-s − 1.47·37-s − 0.320·39-s + 0.624·41-s + 0.304·43-s + 1.16·47-s + 18/7·49-s + 0.280·51-s + 0.824·53-s − 0.662·57-s − 0.781·59-s − 0.256·61-s + 0.610·67-s + 0.722·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.213398067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.213398067\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−9.875639967854719047770408597332, −9.111356026999762054163009438962, −8.804166171198939875992245805865, −8.748884485218176306572166756918, −8.305650469841202826076780632258, −7.85841005260163334362112708333, −7.37057427007622930809697825918, −7.15585286141347379330862059751, −6.52099751971930575338382817385, −6.26673988679016311600048765649, −5.33387325601043710825367660893, −5.26073562020412934192850412699, −4.57961488237064975980004081736, −4.47095343305389669134870380727, −3.82944126478041985109737217208, −3.15499174574357695699034171193, −2.56713018220991306732993624513, −2.21465612184689183522449626548, −1.36397679093702591199882568580, −0.938752316269832272694602333017,
0.938752316269832272694602333017, 1.36397679093702591199882568580, 2.21465612184689183522449626548, 2.56713018220991306732993624513, 3.15499174574357695699034171193, 3.82944126478041985109737217208, 4.47095343305389669134870380727, 4.57961488237064975980004081736, 5.26073562020412934192850412699, 5.33387325601043710825367660893, 6.26673988679016311600048765649, 6.52099751971930575338382817385, 7.15585286141347379330862059751, 7.37057427007622930809697825918, 7.85841005260163334362112708333, 8.305650469841202826076780632258, 8.748884485218176306572166756918, 8.804166171198939875992245805865, 9.111356026999762054163009438962, 9.875639967854719047770408597332
