| L(s) = 1 | − 60·5-s + 980·9-s + 1.78e4·11-s + 6.36e4·19-s + 4.48e4·25-s − 1.69e5·29-s + 3.94e5·31-s + 2.32e5·41-s − 5.88e4·45-s + 2.90e6·49-s − 1.06e6·55-s + 2.09e6·59-s + 5.25e6·61-s + 7.83e6·71-s − 7.72e6·79-s + 8.75e5·81-s − 3.34e7·89-s − 3.81e6·95-s + 1.74e7·99-s − 2.42e7·101-s − 2.15e6·109-s + 1.21e8·121-s − 9.85e6·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.214·5-s + 0.448·9-s + 4.03·11-s + 2.12·19-s + 0.574·25-s − 1.29·29-s + 2.37·31-s + 0.526·41-s − 0.0961·45-s + 3.53·49-s − 0.865·55-s + 1.32·59-s + 2.96·61-s + 2.59·71-s − 1.76·79-s + 0.183·81-s − 5.03·89-s − 0.456·95-s + 1.80·99-s − 2.34·101-s − 0.159·109-s + 6.25·121-s − 0.451·125-s + 0.277·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(22.43541565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.43541565\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 12 p T - 66 p^{4} T^{2} + 12 p^{8} T^{3} + p^{14} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 980 T^{2} + 9382 p^{2} T^{4} - 980 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2907140 T^{2} + 3444026008998 T^{4} - 2907140 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8904 T + 58001046 T^{2} - 8904 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 207134260 T^{2} + 18369334894869078 T^{4} - 207134260 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 513791940 T^{2} + 236061419749305158 T^{4} - 513791940 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 31800 T + 833487878 T^{2} - 31800 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6583044420 T^{2} + 22546653652650262118 T^{4} - 6583044420 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 84780 T + 15469426318 T^{2} + 84780 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 197056 T + 59904732606 T^{2} - 197056 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 238521132500 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - 238521132500 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 116244 T + 205827060246 T^{2} - 116244 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1046615537780 T^{2} + \)\(42\!\cdots\!98\)\( T^{4} - 1046615537780 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 115388894940 T^{2} - \)\(27\!\cdots\!62\)\( T^{4} + 115388894940 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1677817211540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - 1677817211540 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 1046760 T + 4817827968438 T^{2} - 1046760 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 2625716 T + 8002437582606 T^{2} - 2625716 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16580251270100 T^{2} + \)\(13\!\cdots\!58\)\( T^{4} - 16580251270100 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3916176 T + 13255881578926 T^{2} - 3916176 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 37988250868580 T^{2} + \)\(59\!\cdots\!18\)\( T^{4} - 37988250868580 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 3863520 T + 34443978644318 T^{2} + 3863520 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 7805733448980 T^{2} + \)\(31\!\cdots\!58\)\( T^{4} - 7805733448980 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16735020 T + 154740910351158 T^{2} + 16735020 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 161931097215620 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 161931097215620 p^{14} T^{6} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.11743374160044789437628319602, −7.03994711080581731908456989445, −6.61970819378988674545849630815, −6.54622343668685934973996834274, −6.49770087695684684415729710623, −5.62700455071053099945984809452, −5.61741548979769679226506693554, −5.51415976898761875404783157469, −5.38107157662990109691057549271, −4.63026008000880946116307414342, −4.28395307045241891868876005493, −4.25968970670758923267738351226, −3.94354485115344401050257979748, −3.86888037355544621290872931336, −3.58891093678567826642657072721, −3.05680423702925978953488880509, −2.79564909708045133689994042074, −2.63805890681710981305660512848, −1.93984447170844174055953542202, −1.85231941538582846088161960007, −1.32714013589441710935654610596, −1.08962818391827403498534735861, −0.961845713744679897653751052412, −0.73068083463226231619588790042, −0.44832960243000669625195613030,
0.44832960243000669625195613030, 0.73068083463226231619588790042, 0.961845713744679897653751052412, 1.08962818391827403498534735861, 1.32714013589441710935654610596, 1.85231941538582846088161960007, 1.93984447170844174055953542202, 2.63805890681710981305660512848, 2.79564909708045133689994042074, 3.05680423702925978953488880509, 3.58891093678567826642657072721, 3.86888037355544621290872931336, 3.94354485115344401050257979748, 4.25968970670758923267738351226, 4.28395307045241891868876005493, 4.63026008000880946116307414342, 5.38107157662990109691057549271, 5.51415976898761875404783157469, 5.61741548979769679226506693554, 5.62700455071053099945984809452, 6.49770087695684684415729710623, 6.54622343668685934973996834274, 6.61970819378988674545849630815, 7.03994711080581731908456989445, 7.11743374160044789437628319602
