| L(s) = 1 | − 6·7-s − 2·9-s − 6·17-s + 16·23-s − 9·25-s + 8·31-s + 20·41-s − 2·47-s + 13·49-s + 12·63-s + 4·71-s + 18·73-s + 16·79-s − 5·81-s + 24·89-s − 16·97-s − 12·103-s − 20·113-s + 36·119-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 2/3·9-s − 1.45·17-s + 3.33·23-s − 9/5·25-s + 1.43·31-s + 3.12·41-s − 0.291·47-s + 13/7·49-s + 1.51·63-s + 0.474·71-s + 2.10·73-s + 1.80·79-s − 5/9·81-s + 2.54·89-s − 1.62·97-s − 1.18·103-s − 1.88·113-s + 3.30·119-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8718852981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8718852981\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−9.553328544165326093753901438762, −9.084561409834522310688102275122, −9.038018434905016884851105347469, −8.121865683855346250917871835791, −7.62429229152375892340957919668, −6.89999375237673531400272497166, −6.44323514092267885981017350010, −6.35171638937504994192666881666, −5.55162300458149510222680140561, −4.93085486739421392203871722376, −4.15537276955970084853193543262, −3.51647821375355615052320307776, −2.82511261070703297705797963037, −2.48496537170117614857619865014, −0.67525660115678139253233977173,
0.67525660115678139253233977173, 2.48496537170117614857619865014, 2.82511261070703297705797963037, 3.51647821375355615052320307776, 4.15537276955970084853193543262, 4.93085486739421392203871722376, 5.55162300458149510222680140561, 6.35171638937504994192666881666, 6.44323514092267885981017350010, 6.89999375237673531400272497166, 7.62429229152375892340957919668, 8.121865683855346250917871835791, 9.038018434905016884851105347469, 9.084561409834522310688102275122, 9.553328544165326093753901438762
