| L(s) = 1 | − 7-s − 2·13-s + 19-s − 25-s − 2·31-s − 2·37-s + 43-s − 2·61-s + 4·67-s + 73-s + 4·79-s + 2·91-s + 97-s − 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
| L(s) = 1 | − 7-s − 2·13-s + 19-s − 25-s − 2·31-s − 2·37-s + 43-s − 2·61-s + 4·67-s + 73-s + 4·79-s + 2·91-s + 97-s − 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6777293293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6777293293\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.388152575347637883517687112718, −9.106506408278561392699201247540, −8.908253215641278667539417307322, −7.905706890169707268367623280753, −7.87012152210215705770148037674, −7.62974951133162645292250236582, −6.95282322104477309159354470456, −6.75521607995530625804622122707, −6.50257354702596145581126262931, −5.73992924070072981030773410551, −5.38417474670869195490618224536, −5.14793573013088712172720722882, −4.77685309444115870662052277471, −3.94483216189817510174041837858, −3.67709149001288713055609250933, −3.30775238875518169290618052046, −2.66977540780856329821354567858, −2.16374952905319969606544242252, −1.76171378308491608254147435682, −0.54158261353856604625540368453,
0.54158261353856604625540368453, 1.76171378308491608254147435682, 2.16374952905319969606544242252, 2.66977540780856329821354567858, 3.30775238875518169290618052046, 3.67709149001288713055609250933, 3.94483216189817510174041837858, 4.77685309444115870662052277471, 5.14793573013088712172720722882, 5.38417474670869195490618224536, 5.73992924070072981030773410551, 6.50257354702596145581126262931, 6.75521607995530625804622122707, 6.95282322104477309159354470456, 7.62974951133162645292250236582, 7.87012152210215705770148037674, 7.905706890169707268367623280753, 8.908253215641278667539417307322, 9.106506408278561392699201247540, 9.388152575347637883517687112718
