| L(s) = 1 | + 4·2-s + 2·4-s − 20·8-s − 8·9-s + 4·11-s − 45·16-s − 32·18-s + 16·22-s − 12·23-s + 4·25-s − 16·29-s + 16·32-s − 16·36-s − 8·37-s + 12·43-s + 8·44-s − 48·46-s + 16·50-s − 12·53-s − 64·58-s + 204·64-s + 4·67-s + 24·71-s + 160·72-s − 32·74-s − 28·79-s + 30·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4-s − 7.07·8-s − 8/3·9-s + 1.20·11-s − 11.2·16-s − 7.54·18-s + 3.41·22-s − 2.50·23-s + 4/5·25-s − 2.97·29-s + 2.82·32-s − 8/3·36-s − 1.31·37-s + 1.82·43-s + 1.20·44-s − 7.07·46-s + 2.26·50-s − 1.64·53-s − 8.40·58-s + 51/2·64-s + 0.488·67-s + 2.84·71-s + 18.8·72-s − 3.71·74-s − 3.15·79-s + 10/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 31 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 695 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 60 T^{2} + 2399 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T + 72 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 2674 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 172 T^{2} + 13711 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 112 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 127 p T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 14434 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.89684027665791751504363136249, −5.49794841782714958544543515038, −5.38387949042911495030573417009, −5.35818273727973063038988670571, −5.12381173945497185200715969103, −4.86858001278640850079162306416, −4.78551870415636199723275737793, −4.67744596040200875337859874821, −4.27252186713279193301829993121, −3.97572472596881026942848007382, −3.95060620832981821275479969695, −3.93739922237364483028673655459, −3.72581325276870932629292132936, −3.58309319328120271482630293546, −3.38127624419801894077834048120, −3.32558283982703256973298074961, −2.77653583219558343960127506953, −2.61097585469625325153175250221, −2.58735428831980365958663128867, −2.56152764397577830063541114858, −2.19595322878054087625606661067, −1.66724417693147768874489687637, −1.28233310016117794014915772548, −1.26674383787486553216215737711, −0.71557325516128885097842539126, 0, 0, 0, 0,
0.71557325516128885097842539126, 1.26674383787486553216215737711, 1.28233310016117794014915772548, 1.66724417693147768874489687637, 2.19595322878054087625606661067, 2.56152764397577830063541114858, 2.58735428831980365958663128867, 2.61097585469625325153175250221, 2.77653583219558343960127506953, 3.32558283982703256973298074961, 3.38127624419801894077834048120, 3.58309319328120271482630293546, 3.72581325276870932629292132936, 3.93739922237364483028673655459, 3.95060620832981821275479969695, 3.97572472596881026942848007382, 4.27252186713279193301829993121, 4.67744596040200875337859874821, 4.78551870415636199723275737793, 4.86858001278640850079162306416, 5.12381173945497185200715969103, 5.35818273727973063038988670571, 5.38387949042911495030573417009, 5.49794841782714958544543515038, 5.89684027665791751504363136249
