| L(s) = 1 | − 8·4-s + 100·9-s + 44·11-s + 48·16-s + 96·19-s + 106·25-s + 224·29-s − 560·31-s − 800·36-s − 944·41-s − 352·44-s + 616·49-s − 1.79e3·59-s − 168·61-s − 256·64-s + 3.36e3·71-s − 768·76-s − 2.20e3·79-s + 6.04e3·81-s − 1.71e3·89-s + 4.40e3·99-s − 848·100-s + 3.07e3·101-s − 152·109-s − 1.79e3·116-s + 1.21e3·121-s + 4.48e3·124-s + ⋯ |
| L(s) = 1 | − 4-s + 3.70·9-s + 1.20·11-s + 3/4·16-s + 1.15·19-s + 0.847·25-s + 1.43·29-s − 3.24·31-s − 3.70·36-s − 3.59·41-s − 1.20·44-s + 1.79·49-s − 3.95·59-s − 0.352·61-s − 1/2·64-s + 5.61·71-s − 1.15·76-s − 3.13·79-s + 8.28·81-s − 2.03·89-s + 4.46·99-s − 0.847·100-s + 3.02·101-s − 0.133·109-s − 1.43·116-s + 0.909·121-s + 3.24·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.004700956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.004700956\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 106 T^{2} + p^{6} T^{4} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 88 p T^{2} + 227278 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 8799118 T^{4} + 32 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6880 T^{2} + 24784638 T^{4} - 6880 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 48 T + 1478 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36020 T^{2} + 592521478 T^{4} - 36020 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 112 T + 8838 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 280 T + 77758 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 120740 T^{2} + 7380327318 T^{4} - 120740 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 472 T + 133374 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 219928 T^{2} + 23196089598 T^{4} - 219928 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 249332 T^{2} + 35339203558 T^{4} - 249332 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 337828 T^{2} + 72694104918 T^{4} - 337828 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 896 T + 610038 T^{2} + 896 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 84 T - 5650 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1122340 T^{2} + 495771573238 T^{4} - 1122340 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1680 T + 1398638 T^{2} - 1680 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 406528 T^{2} + 244592823838 T^{4} - 406528 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1100 T + 1208478 T^{2} + 1100 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 152360 T^{2} + 271647619038 T^{4} + 152360 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 856 T + 806718 T^{2} + 856 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1176052 T^{2} + 91240422 p^{2} T^{4} - 1176052 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−9.800734437335247691397863474405, −9.157893254319049217808694363441, −9.152581604099931875761672567245, −8.943970589583339477675317732706, −8.433702023892961038269012616632, −8.180451811350527063414792829440, −7.68519611909691653166333082116, −7.36847755071267905517250640597, −7.30020527482840485129820830534, −6.85919156798442937645170079746, −6.72984782501044582968569266444, −6.50906188396059807434152747549, −5.78444138239734834325603960422, −5.56184667926868210031524018254, −4.79905540395100651111274932447, −4.78143493148932293785358250162, −4.75417019382572993064376705297, −3.93114210041686581600781635461, −3.85962947244232502949673496585, −3.52527549463461634882421642086, −3.07959922954905578621463451378, −1.87707382217377094110812866789, −1.65924788761949751264173222382, −1.26287969055886934657329879926, −0.63154234381133739842943529921,
0.63154234381133739842943529921, 1.26287969055886934657329879926, 1.65924788761949751264173222382, 1.87707382217377094110812866789, 3.07959922954905578621463451378, 3.52527549463461634882421642086, 3.85962947244232502949673496585, 3.93114210041686581600781635461, 4.75417019382572993064376705297, 4.78143493148932293785358250162, 4.79905540395100651111274932447, 5.56184667926868210031524018254, 5.78444138239734834325603960422, 6.50906188396059807434152747549, 6.72984782501044582968569266444, 6.85919156798442937645170079746, 7.30020527482840485129820830534, 7.36847755071267905517250640597, 7.68519611909691653166333082116, 8.180451811350527063414792829440, 8.433702023892961038269012616632, 8.943970589583339477675317732706, 9.152581604099931875761672567245, 9.157893254319049217808694363441, 9.800734437335247691397863474405
