| L(s) = 1 | − 2·9-s − 68·13-s − 228·17-s − 52·29-s + 300·37-s + 684·41-s − 634·49-s + 524·53-s − 524·61-s − 1.36e3·73-s − 725·81-s − 1.26e3·89-s + 1.93e3·97-s + 3.27e3·101-s − 684·109-s − 4.21e3·113-s + 136·117-s − 790·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 456·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 0.0740·9-s − 1.45·13-s − 3.25·17-s − 0.332·29-s + 1.33·37-s + 2.60·41-s − 1.84·49-s + 1.35·53-s − 1.09·61-s − 2.18·73-s − 0.994·81-s − 1.50·89-s + 2.02·97-s + 3.22·101-s − 0.601·109-s − 3.50·113-s + 0.107·117-s − 0.593·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.240·153-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 634 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 790 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 114 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19398 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 49390 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 342 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 47374 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 133526 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 262 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 170310 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 353954 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 392610 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 682 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 945310 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1120642 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 966 T + p^{3} T^{2} )^{2} \) |
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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
−9.435321919249847611206945803769, −9.343344605530336337508744796049, −8.882006751474385295082998828826, −8.506765984973223994049238319744, −7.891972398216309633844737399447, −7.38921702064259480566139109962, −7.20455728540037458276011241102, −6.60949343588070955874911551668, −6.09845100433318138500483797778, −5.90157241197174942214725569033, −4.91986499002680235928212550856, −4.72729778786927424373666589601, −4.28275592729231520457663884859, −3.83773918354454230414906100058, −2.76397405131526570562379845970, −2.53360295444243716703651142835, −2.07518152857865185989793599963, −1.16741478139356033429690333780, 0, 0,
1.16741478139356033429690333780, 2.07518152857865185989793599963, 2.53360295444243716703651142835, 2.76397405131526570562379845970, 3.83773918354454230414906100058, 4.28275592729231520457663884859, 4.72729778786927424373666589601, 4.91986499002680235928212550856, 5.90157241197174942214725569033, 6.09845100433318138500483797778, 6.60949343588070955874911551668, 7.20455728540037458276011241102, 7.38921702064259480566139109962, 7.891972398216309633844737399447, 8.506765984973223994049238319744, 8.882006751474385295082998828826, 9.343344605530336337508744796049, 9.435321919249847611206945803769
