| 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
| \(L(4)\) |
\(\approx\) |
\(0.8433961726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8433961726\) |
| \(L(\frac{9}{2})\) |
|
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
−7.25758092255765768609577627219, −7.21979519680422270901784629002, −6.74771461130867415790779857777, −6.51135682545239918479401858808, −6.01629066780093993710600091598, −5.69971886014494970687170927086, −5.63078500902529769050172763587, −5.33369864801132042140414882814, −5.28866452574844472383159842541, −4.96069016852743988143334079534, −4.36827090603305120005551552591, −4.35569488307931930659742858325, −4.07120528671357125548558034250, −3.61386371896259072174770603504, −3.58130872356460692346206504124, −2.76516714365339954124279984486, −2.64383506660531027113804264882, −2.62700045696566146019322052304, −2.34274339097529629546039661020, −1.80839974866075552987691784820, −1.69923294251145476340091721914, −1.23706031847505651707204480756, −0.59238040071224973957025445004, −0.33256610947160050795208606240, −0.20909853438736715559702609899,
0.20909853438736715559702609899, 0.33256610947160050795208606240, 0.59238040071224973957025445004, 1.23706031847505651707204480756, 1.69923294251145476340091721914, 1.80839974866075552987691784820, 2.34274339097529629546039661020, 2.62700045696566146019322052304, 2.64383506660531027113804264882, 2.76516714365339954124279984486, 3.58130872356460692346206504124, 3.61386371896259072174770603504, 4.07120528671357125548558034250, 4.35569488307931930659742858325, 4.36827090603305120005551552591, 4.96069016852743988143334079534, 5.28866452574844472383159842541, 5.33369864801132042140414882814, 5.63078500902529769050172763587, 5.69971886014494970687170927086, 6.01629066780093993710600091598, 6.51135682545239918479401858808, 6.74771461130867415790779857777, 7.21979519680422270901784629002, 7.25758092255765768609577627219
