| L(s) = 1 | + 2·2-s + 8·3-s − 24·4-s + 38·5-s + 16·6-s − 40·8-s − 401·9-s + 76·10-s + 424·11-s − 192·12-s + 924·13-s + 304·15-s − 304·16-s + 2.34e3·17-s − 802·18-s + 360·19-s − 912·20-s + 848·22-s − 12·23-s − 320·24-s − 1.46e3·25-s + 1.84e3·26-s − 4.98e3·27-s − 7.05e3·29-s + 608·30-s − 3.54e3·31-s − 3.07e3·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.513·3-s − 3/4·4-s + 0.679·5-s + 0.181·6-s − 0.220·8-s − 1.65·9-s + 0.240·10-s + 1.05·11-s − 0.384·12-s + 1.51·13-s + 0.348·15-s − 0.296·16-s + 1.96·17-s − 0.583·18-s + 0.228·19-s − 0.509·20-s + 0.373·22-s − 0.00473·23-s − 0.113·24-s − 0.469·25-s + 0.536·26-s − 1.31·27-s − 1.55·29-s + 0.123·30-s − 0.663·31-s − 0.530·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.863849664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.863849664\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 7 p^{2} T^{2} - p^{6} T^{3} + p^{10} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 8 T + 155 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 38 T + 2911 T^{2} - 38 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 424 T + 347473 T^{2} - 424 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 924 T + 927022 T^{2} - 924 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 138 p T + 3570955 T^{2} - 138 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 360 T + 2145625 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 12696565 T^{2} + 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7052 T + 35324974 T^{2} + 7052 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3548 T + 41490053 T^{2} + 3548 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11090 T + 146343239 T^{2} - 11090 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3500 T + 206898214 T^{2} + 3500 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12680 T + 267378054 T^{2} + 12680 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 22956 T + 491525173 T^{2} - 22956 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3042 T + 716414839 T^{2} - 3042 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 65808 T + 2502089257 T^{2} - 65808 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 42486 T + 1501996159 T^{2} - 42486 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 42312 T + 3116577793 T^{2} - 42312 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2208 T + 3433192846 T^{2} + 2208 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 50506 T + 2773040987 T^{2} - 50506 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9004 T + 5176592589 T^{2} - 9004 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 104328 T + 9837878230 T^{2} - 104328 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 26666 T + 8107102555 T^{2} - 26666 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2156 p T + 28107307478 T^{2} + 2156 p^{6} T^{3} + p^{10} T^{4} \) |
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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
−14.60190050376410799364368116107, −14.31878321429291714800765399999, −13.64913374811210314858378195923, −13.55297468579529415140003088813, −12.79297713179705541576307453515, −11.96408955938613676192697160370, −11.35977846119517956365892102443, −10.99283032463246233433346104171, −9.648223823811190284022303785448, −9.639027641788965207130678626217, −8.679442646246045687763180557989, −8.458612766040171374972464918030, −7.55012113964279976394725156723, −6.42517346785260233871620965577, −5.54579878856861627220168284157, −5.48944517134452957758436702474, −3.71738055932699853720200953201, −3.63234521411195907509385583631, −2.18066979147607435516174234252, −0.855700788601722691022264992956,
0.855700788601722691022264992956, 2.18066979147607435516174234252, 3.63234521411195907509385583631, 3.71738055932699853720200953201, 5.48944517134452957758436702474, 5.54579878856861627220168284157, 6.42517346785260233871620965577, 7.55012113964279976394725156723, 8.458612766040171374972464918030, 8.679442646246045687763180557989, 9.639027641788965207130678626217, 9.648223823811190284022303785448, 10.99283032463246233433346104171, 11.35977846119517956365892102443, 11.96408955938613676192697160370, 12.79297713179705541576307453515, 13.55297468579529415140003088813, 13.64913374811210314858378195923, 14.31878321429291714800765399999, 14.60190050376410799364368116107
