| L(s) = 1 | − 2·4-s − 5-s − 4·9-s + 3·11-s + 5·19-s + 2·20-s − 4·25-s + 11·29-s − 6·31-s + 8·36-s − 41-s − 6·44-s + 4·45-s − 6·49-s − 3·55-s + 59-s − 2·61-s + 8·64-s − 3·71-s − 10·76-s − 15·79-s + 7·81-s − 9·89-s − 5·95-s − 12·99-s + 8·100-s + 15·101-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 4/3·9-s + 0.904·11-s + 1.14·19-s + 0.447·20-s − 4/5·25-s + 2.04·29-s − 1.07·31-s + 4/3·36-s − 0.156·41-s − 0.904·44-s + 0.596·45-s − 6/7·49-s − 0.404·55-s + 0.130·59-s − 0.256·61-s + 64-s − 0.356·71-s − 1.14·76-s − 1.68·79-s + 7/9·81-s − 0.953·89-s − 0.512·95-s − 1.20·99-s + 4/5·100-s + 1.49·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4222141590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4222141590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 109 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 152 T^{2} + p^{2} T^{4} \) |
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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
−14.19454045717374451127091353927, −13.78836223859467724349513577428, −13.02132215935876782760497681759, −12.16321002560880394673395714560, −11.64675702466879730087811532096, −11.19128473596752981447430276521, −10.09738201918885273142298867748, −9.423661061919065844869674749372, −8.748731100307867456017587191214, −8.282454962036726379628037425237, −7.29437775775195108905884201252, −6.23530102487327827298160382161, −5.31271891550307177711333795025, −4.34036075847737870492735412284, −3.22692316031770542346454550404,
3.22692316031770542346454550404, 4.34036075847737870492735412284, 5.31271891550307177711333795025, 6.23530102487327827298160382161, 7.29437775775195108905884201252, 8.282454962036726379628037425237, 8.748731100307867456017587191214, 9.423661061919065844869674749372, 10.09738201918885273142298867748, 11.19128473596752981447430276521, 11.64675702466879730087811532096, 12.16321002560880394673395714560, 13.02132215935876782760497681759, 13.78836223859467724349513577428, 14.19454045717374451127091353927
