| L(s) = 1 | − 4-s − 5-s − 3·7-s − 2·9-s + 5·11-s − 3·13-s + 16-s + 17-s + 19-s + 20-s + 2·23-s − 2·25-s + 3·27-s + 3·28-s − 29-s + 2·31-s + 3·35-s + 2·36-s − 10·37-s + 9·41-s + 4·43-s − 5·44-s + 2·45-s − 8·47-s − 2·49-s + 3·52-s + 2·53-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.447·5-s − 1.13·7-s − 2/3·9-s + 1.50·11-s − 0.832·13-s + 1/4·16-s + 0.242·17-s + 0.229·19-s + 0.223·20-s + 0.417·23-s − 2/5·25-s + 0.577·27-s + 0.566·28-s − 0.185·29-s + 0.359·31-s + 0.507·35-s + 1/3·36-s − 1.64·37-s + 1.40·41-s + 0.609·43-s − 0.753·44-s + 0.298·45-s − 1.16·47-s − 2/7·49-s + 0.416·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1164 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1164 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4560847758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4560847758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 17 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - T - 37 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 101 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−19.6336328981, −19.2751361086, −18.9675421624, −17.9229113069, −17.5269618026, −16.9541438335, −16.4348010092, −15.9479729115, −15.1012691382, −14.6001545438, −14.0459878331, −13.5162536258, −12.5219853119, −12.3073374844, −11.5959848517, −10.8526783656, −9.87134223281, −9.41650093466, −8.82770492731, −7.95016706298, −6.97260843625, −6.33610405553, −5.28150323691, −4.10178342402, −3.15232834929,
3.15232834929, 4.10178342402, 5.28150323691, 6.33610405553, 6.97260843625, 7.95016706298, 8.82770492731, 9.41650093466, 9.87134223281, 10.8526783656, 11.5959848517, 12.3073374844, 12.5219853119, 13.5162536258, 14.0459878331, 14.6001545438, 15.1012691382, 15.9479729115, 16.4348010092, 16.9541438335, 17.5269618026, 17.9229113069, 18.9675421624, 19.2751361086, 19.6336328981
