| L(s) = 1 | − 4·4-s − 9·9-s − 96·11-s + 16·16-s − 280·19-s − 420·29-s + 544·31-s + 36·36-s − 396·41-s + 384·44-s + 670·49-s − 480·59-s + 604·61-s − 64·64-s − 1.53e3·71-s + 1.12e3·76-s + 1.28e3·79-s + 81·81-s − 420·89-s + 864·99-s + 3.44e3·101-s + 1.22e3·109-s + 1.68e3·116-s + 4.25e3·121-s − 2.17e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.63·11-s + 1/4·16-s − 3.38·19-s − 2.68·29-s + 3.15·31-s + 1/6·36-s − 1.50·41-s + 1.31·44-s + 1.95·49-s − 1.05·59-s + 1.26·61-s − 1/8·64-s − 2.56·71-s + 1.69·76-s + 1.82·79-s + 1/9·81-s − 0.500·89-s + 0.877·99-s + 3.39·101-s + 1.07·109-s + 1.34·116-s + 3.19·121-s − 1.57·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3506597128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3506597128\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4390 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3170 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 272 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 10250 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 87190 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 160990 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291670 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 302 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 246310 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 549550 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 640 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1022470 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 527810 T^{2} + p^{6} T^{4} \) |
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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
−12.86979543112994399977079057624, −12.56129776480021685096843606329, −11.78821021563569664780997133751, −11.21083133443429788769480921893, −10.59147494896833370313373735651, −10.22576905109508936003417735427, −10.12795762288647170570320878774, −8.823280012398518663406337505394, −8.815830045508566178127693547766, −8.008723332242600693263282145555, −7.82624810855910269559865874379, −6.96407135356038098304008064783, −6.13617407268324372371099619580, −5.76280917360728651188354811322, −4.89803569275988450126733203883, −4.54470384811990768391320096075, −3.64809476994345000178108283069, −2.58079993469889873831825000244, −2.15063714118031055240266891461, −0.26970457103126915749408202189,
0.26970457103126915749408202189, 2.15063714118031055240266891461, 2.58079993469889873831825000244, 3.64809476994345000178108283069, 4.54470384811990768391320096075, 4.89803569275988450126733203883, 5.76280917360728651188354811322, 6.13617407268324372371099619580, 6.96407135356038098304008064783, 7.82624810855910269559865874379, 8.008723332242600693263282145555, 8.815830045508566178127693547766, 8.823280012398518663406337505394, 10.12795762288647170570320878774, 10.22576905109508936003417735427, 10.59147494896833370313373735651, 11.21083133443429788769480921893, 11.78821021563569664780997133751, 12.56129776480021685096843606329, 12.86979543112994399977079057624
