| L(s) = 1 | − 4·5-s − 4·7-s − 16·11-s − 32·13-s − 40·17-s − 56·23-s + 16·25-s + 16·31-s + 16·35-s + 64·37-s − 56·41-s + 8·43-s − 128·47-s + 8·49-s + 56·53-s + 64·55-s + 200·61-s + 128·65-s + 200·67-s + 272·71-s + 76·73-s + 64·77-s − 9·81-s + 16·83-s + 160·85-s + 128·91-s − 20·97-s + ⋯ |
| L(s) = 1 | − 4/5·5-s − 4/7·7-s − 1.45·11-s − 2.46·13-s − 2.35·17-s − 2.43·23-s + 0.639·25-s + 0.516·31-s + 0.457·35-s + 1.72·37-s − 1.36·41-s + 8/43·43-s − 2.72·47-s + 8/49·49-s + 1.05·53-s + 1.16·55-s + 3.27·61-s + 1.96·65-s + 2.98·67-s + 3.83·71-s + 1.04·73-s + 0.831·77-s − 1/9·81-s + 0.192·83-s + 1.88·85-s + 1.40·91-s − 0.206·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1626357947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1626357947\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 156 T^{3} + 2942 T^{4} + 156 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 204 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 9120 T^{3} + 148994 T^{4} + 9120 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 15240 T^{3} + 281858 T^{4} + 15240 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} + 50904 T^{3} + 1508162 T^{4} + 50904 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1722 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 64 T + 2048 T^{2} - 58176 T^{3} + 1440962 T^{4} - 58176 p^{2} T^{5} + 2048 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 28 T + 3342 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 5256 T^{3} - 557566 T^{4} - 5256 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 128 T + 8192 T^{2} + 506496 T^{3} + 28260194 T^{4} + 506496 p^{2} T^{5} + 8192 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 155064 T^{3} + 15333122 T^{4} - 155064 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 100 T + 7998 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 1888200 T^{3} + 153742658 T^{4} - 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T + 2888 T^{2} + 65436 T^{3} - 36833458 T^{4} + 65436 p^{2} T^{5} + 2888 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 11882 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 101328 T^{3} + 79904642 T^{4} - 101328 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 173820 T^{3} + 150551438 T^{4} + 173820 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.368092775420295405235965785671, −8.323685159176518791250024398131, −8.292774113657205071105400873135, −7.78046511787214570316113629392, −7.64847012972257220275329986923, −7.29451298350207619045197711856, −6.98709449270415728515025754828, −6.82705523775956883503006196835, −6.45615254347630906647724536268, −6.24157634533574064888925930521, −6.12679795728778196076819414974, −5.38049108409829782976410266800, −5.03254301052183749012983187564, −4.98019756758232554948747394497, −4.92694710288158131367275526607, −4.41780163088280891667467582333, −3.83087607854624593658669646398, −3.71164184470826098600082485207, −3.60130278412735954220263941563, −2.51665413217992251815533356522, −2.51138712799957098980902801806, −2.36466950258511962793369091954, −2.01644564885115703703541077465, −0.72005703561967498738586511131, −0.14765568633595761737213549879,
0.14765568633595761737213549879, 0.72005703561967498738586511131, 2.01644564885115703703541077465, 2.36466950258511962793369091954, 2.51138712799957098980902801806, 2.51665413217992251815533356522, 3.60130278412735954220263941563, 3.71164184470826098600082485207, 3.83087607854624593658669646398, 4.41780163088280891667467582333, 4.92694710288158131367275526607, 4.98019756758232554948747394497, 5.03254301052183749012983187564, 5.38049108409829782976410266800, 6.12679795728778196076819414974, 6.24157634533574064888925930521, 6.45615254347630906647724536268, 6.82705523775956883503006196835, 6.98709449270415728515025754828, 7.29451298350207619045197711856, 7.64847012972257220275329986923, 7.78046511787214570316113629392, 8.292774113657205071105400873135, 8.323685159176518791250024398131, 8.368092775420295405235965785671
