| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 4·14-s + 16-s − 4·17-s − 2·23-s + 6·25-s − 4·28-s − 7·31-s + 32-s − 4·34-s + 4·41-s − 2·46-s + 16·47-s − 2·49-s + 6·50-s − 4·56-s − 7·62-s + 64-s − 4·68-s + 4·71-s + 18·73-s − 9·81-s + 4·82-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.417·23-s + 6/5·25-s − 0.755·28-s − 1.25·31-s + 0.176·32-s − 0.685·34-s + 0.624·41-s − 0.294·46-s + 2.33·47-s − 2/7·49-s + 0.848·50-s − 0.534·56-s − 0.889·62-s + 1/8·64-s − 0.485·68-s + 0.474·71-s + 2.10·73-s − 81-s + 0.441·82-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015304061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015304061\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−12.53487890234502064443148663707, −12.24560156237213685970206944839, −11.29061985006174004231665785142, −10.82806256412385088591450350359, −10.24781718258600588205761760019, −9.409632741766434253895968277129, −9.085016816626631737772979658916, −8.159481626222407720241007792756, −7.20109961112149975417613119511, −6.72791953092319215897878431738, −6.09145757988806094537072087683, −5.31755384946885081122973710446, −4.28343417363558192169286993651, −3.47976003890724965430926963307, −2.49646319992430295953231830894,
2.49646319992430295953231830894, 3.47976003890724965430926963307, 4.28343417363558192169286993651, 5.31755384946885081122973710446, 6.09145757988806094537072087683, 6.72791953092319215897878431738, 7.20109961112149975417613119511, 8.159481626222407720241007792756, 9.085016816626631737772979658916, 9.409632741766434253895968277129, 10.24781718258600588205761760019, 10.82806256412385088591450350359, 11.29061985006174004231665785142, 12.24560156237213685970206944839, 12.53487890234502064443148663707
