| L(s) = 1 | + 4-s − 3·9-s + 16-s − 12·17-s + 8·19-s + 16·23-s − 6·25-s + 4·29-s − 31-s − 3·36-s − 14·49-s − 12·53-s + 64-s − 24·67-s − 12·68-s + 8·76-s + 9·81-s + 16·83-s − 12·89-s + 16·92-s + 4·97-s − 6·100-s + 16·103-s − 4·109-s + 4·116-s − 22·121-s − 124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 9-s + 1/4·16-s − 2.91·17-s + 1.83·19-s + 3.33·23-s − 6/5·25-s + 0.742·29-s − 0.179·31-s − 1/2·36-s − 2·49-s − 1.64·53-s + 1/8·64-s − 2.93·67-s − 1.45·68-s + 0.917·76-s + 81-s + 1.75·83-s − 1.27·89-s + 1.66·92-s + 0.406·97-s − 3/5·100-s + 1.57·103-s − 0.383·109-s + 0.371·116-s − 2·121-s − 0.0898·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1072476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1072476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 31 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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
−7.76857292337562945056886541715, −7.43556895569719724485579804346, −6.93807340940953532824726950400, −6.48842833802306821880895380523, −6.32858579420662158347624019611, −5.61110203252813853267502993088, −5.11987644824288757943667009350, −4.74203912193695697715499891246, −4.37664321587824825312436918660, −3.29788473612982991726310940500, −3.16188554299090604324575557253, −2.63802468934373486533184492979, −1.92931953630680217049542026766, −1.15997339964107602320819529739, 0,
1.15997339964107602320819529739, 1.92931953630680217049542026766, 2.63802468934373486533184492979, 3.16188554299090604324575557253, 3.29788473612982991726310940500, 4.37664321587824825312436918660, 4.74203912193695697715499891246, 5.11987644824288757943667009350, 5.61110203252813853267502993088, 6.32858579420662158347624019611, 6.48842833802306821880895380523, 6.93807340940953532824726950400, 7.43556895569719724485579804346, 7.76857292337562945056886541715
