| L(s) = 1 | + 4·3-s − 4-s + 10·9-s − 4·12-s + 16-s − 32·23-s + 20·27-s − 2·31-s − 10·36-s − 20·37-s + 12·47-s + 4·48-s − 21·49-s + 28·53-s + 16·59-s − 5·64-s + 10·67-s − 128·69-s + 20·71-s + 35·81-s − 24·89-s + 32·92-s − 8·93-s + 26·97-s + 8·103-s − 20·108-s − 80·111-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1/2·4-s + 10/3·9-s − 1.15·12-s + 1/4·16-s − 6.67·23-s + 3.84·27-s − 0.359·31-s − 5/3·36-s − 3.28·37-s + 1.75·47-s + 0.577·48-s − 3·49-s + 3.84·53-s + 2.08·59-s − 5/8·64-s + 1.22·67-s − 15.4·69-s + 2.37·71-s + 35/9·81-s − 2.54·89-s + 3.33·92-s − 0.829·93-s + 2.63·97-s + 0.788·103-s − 1.92·108-s − 7.59·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9673172168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9673172168\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4$ | \( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 3 p T^{2} + 200 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 6 T^{2} - 181 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T^{2} + 320 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 426 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_4\times C_2$ | \( 1 + 201 T^{2} + 17336 T^{4} + 201 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 5 T + 66 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 134 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 270 T^{2} + 30179 T^{4} + 270 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 256 T^{2} + 28974 T^{4} + 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 13 T + 228 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.54563662159613731142920332301, −5.06550812826629806382288687223, −5.00302867124636698389665951687, −4.88958027758379842975396179651, −4.70398829981743789645496452258, −4.25879365006697737128370813263, −4.03858824748523006369017109223, −3.98871516781246390069405398033, −3.92743727211467745390012188302, −3.77290766002613901238578097720, −3.65371307526976639292889787358, −3.53465375237912974058674869952, −3.21438635880729828609354708176, −2.94595695000723963534878952943, −2.69586312602104485176553011759, −2.32913189735522683948155299874, −2.29312644746199622747525219400, −2.10844432305972024748007763575, −1.98602178282388334004510061181, −1.82689485134780157405556520597, −1.62515702012049602368677664633, −1.16257966360915629108013288306, −0.963665574910324491318257092373, −0.45827720590177441134170960819, −0.098829654736545109562797906734,
0.098829654736545109562797906734, 0.45827720590177441134170960819, 0.963665574910324491318257092373, 1.16257966360915629108013288306, 1.62515702012049602368677664633, 1.82689485134780157405556520597, 1.98602178282388334004510061181, 2.10844432305972024748007763575, 2.29312644746199622747525219400, 2.32913189735522683948155299874, 2.69586312602104485176553011759, 2.94595695000723963534878952943, 3.21438635880729828609354708176, 3.53465375237912974058674869952, 3.65371307526976639292889787358, 3.77290766002613901238578097720, 3.92743727211467745390012188302, 3.98871516781246390069405398033, 4.03858824748523006369017109223, 4.25879365006697737128370813263, 4.70398829981743789645496452258, 4.88958027758379842975396179651, 5.00302867124636698389665951687, 5.06550812826629806382288687223, 5.54563662159613731142920332301
