| L(s) = 1 | + 2·4-s − 4·11-s + 3·16-s + 8·29-s − 24·41-s − 8·44-s + 20·49-s − 16·59-s + 8·61-s + 12·64-s − 16·79-s − 8·89-s + 8·101-s + 8·109-s + 16·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 48·164-s + 167-s + 4·169-s + ⋯ |
| L(s) = 1 | + 4-s − 1.20·11-s + 3/4·16-s + 1.48·29-s − 3.74·41-s − 1.20·44-s + 20/7·49-s − 2.08·59-s + 1.02·61-s + 3/2·64-s − 1.80·79-s − 0.847·89-s + 0.796·101-s + 0.766·109-s + 1.48·116-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 3.74·164-s + 0.0773·167-s + 4/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.291359092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291359092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 170 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2454 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 6486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25030 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.42157375500980571693457986590, −6.30002296784641048810143140823, −5.75927646012269045906764173688, −5.69516050778051913047201212083, −5.66324912172298670937505015561, −5.25464414472577267861072262022, −5.11859164278606947358614517544, −5.03816391174482433433435800034, −4.67929527846158112829302915121, −4.34485160229358618802050751053, −4.29022071614590204580742344669, −4.12980074950403156154230096914, −3.64625957398858070236307501710, −3.31768717707934086109666035062, −3.27936586476421750957615978291, −3.13165290902666212124197552909, −2.85277762450337239895914591725, −2.41680025702599308989805076111, −2.26788405973561448328255368373, −2.17230378619256087587225470884, −1.81932044797704165512034690848, −1.28331792487326790948264980977, −1.26411635372000649165420672362, −0.76304003956806212336008103255, −0.17912477784042206012875641648,
0.17912477784042206012875641648, 0.76304003956806212336008103255, 1.26411635372000649165420672362, 1.28331792487326790948264980977, 1.81932044797704165512034690848, 2.17230378619256087587225470884, 2.26788405973561448328255368373, 2.41680025702599308989805076111, 2.85277762450337239895914591725, 3.13165290902666212124197552909, 3.27936586476421750957615978291, 3.31768717707934086109666035062, 3.64625957398858070236307501710, 4.12980074950403156154230096914, 4.29022071614590204580742344669, 4.34485160229358618802050751053, 4.67929527846158112829302915121, 5.03816391174482433433435800034, 5.11859164278606947358614517544, 5.25464414472577267861072262022, 5.66324912172298670937505015561, 5.69516050778051913047201212083, 5.75927646012269045906764173688, 6.30002296784641048810143140823, 6.42157375500980571693457986590
