| L(s) = 1 | − 4·4-s + 3·5-s − 9-s + 10·16-s − 12·20-s + 5·23-s + 6·25-s + 31-s + 4·36-s − 3·45-s − 3·47-s − 4·49-s + 3·59-s − 20·64-s − 2·67-s + 2·71-s + 30·80-s − 20·92-s + 2·97-s − 24·100-s + 2·103-s − 2·113-s + 15·115-s − 4·124-s + 10·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4·4-s + 3·5-s − 9-s + 10·16-s − 12·20-s + 5·23-s + 6·25-s + 31-s + 4·36-s − 3·45-s − 3·47-s − 4·49-s + 3·59-s − 20·64-s − 2·67-s + 2·71-s + 30·80-s − 20·92-s + 2·97-s − 24·100-s + 2·103-s − 2·113-s + 15·115-s − 4·124-s + 10·125-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.341831786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.341831786\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| 31 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
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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
−6.13063247515123574298012772331, −5.84257416073717315958684290220, −5.73395049661434522570367837059, −5.37581908436124495815416880089, −5.34242272826820922879674468034, −5.06161136115338882233992946851, −5.06019988255498499975118980542, −4.88516118979685727675920366789, −4.85872805893364017976750491253, −4.50968662827010846171187490578, −4.50525607172938350622754800900, −4.12755638025907934968836263638, −3.56042116071381039590998719517, −3.41308810787272541818645657369, −3.41030114179003885484446295539, −3.16932801058873083676924285993, −2.97333894924370551672699495606, −2.69778475351704603663550900617, −2.54507860571260005338407315781, −2.01779512216252698686975304614, −1.58409939556362656583633102958, −1.44185526976509011453019277978, −1.16102536783901361078357512959, −0.904218784242329787342517096829, −0.59611671820850027271645145416,
0.59611671820850027271645145416, 0.904218784242329787342517096829, 1.16102536783901361078357512959, 1.44185526976509011453019277978, 1.58409939556362656583633102958, 2.01779512216252698686975304614, 2.54507860571260005338407315781, 2.69778475351704603663550900617, 2.97333894924370551672699495606, 3.16932801058873083676924285993, 3.41030114179003885484446295539, 3.41308810787272541818645657369, 3.56042116071381039590998719517, 4.12755638025907934968836263638, 4.50525607172938350622754800900, 4.50968662827010846171187490578, 4.85872805893364017976750491253, 4.88516118979685727675920366789, 5.06019988255498499975118980542, 5.06161136115338882233992946851, 5.34242272826820922879674468034, 5.37581908436124495815416880089, 5.73395049661434522570367837059, 5.84257416073717315958684290220, 6.13063247515123574298012772331
