| L(s) = 1 | − 42·3-s + 4·7-s + 1.03e3·9-s + 5.90e3·13-s − 1.05e4·19-s − 168·21-s + 2.45e3·25-s − 1.28e4·27-s + 4.57e4·31-s − 6.81e4·37-s − 2.47e5·39-s + 1.28e4·43-s − 2.35e5·49-s + 4.41e5·57-s + 1.25e5·61-s + 4.14e3·63-s − 8.77e5·67-s − 1.46e6·73-s − 1.02e5·75-s + 6.81e5·79-s − 2.14e5·81-s + 2.36e4·91-s − 1.92e6·93-s − 5.62e5·97-s − 1.73e6·103-s − 1.30e6·109-s + 2.86e6·111-s + ⋯ |
| L(s) = 1 | − 1.55·3-s + 0.0116·7-s + 1.41·9-s + 2.68·13-s − 1.53·19-s − 0.0181·21-s + 0.156·25-s − 0.652·27-s + 1.53·31-s − 1.34·37-s − 4.17·39-s + 0.161·43-s − 1.99·49-s + 2.38·57-s + 0.551·61-s + 0.0165·63-s − 2.91·67-s − 3.75·73-s − 0.243·75-s + 1.38·79-s − 0.404·81-s + 0.0313·91-s − 2.39·93-s − 0.615·97-s − 1.58·103-s − 1.00·109-s + 2.09·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6696458163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6696458163\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + 14 p T + p^{6} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 98 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3541970 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2950 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 28202690 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5258 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 191004770 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1184779442 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 22898 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34058 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9219079010 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6406 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10801249342 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7253988050 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22449655150 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 62566 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 438698 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 251546372642 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 730510 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 340562 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 407613512306 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 844406214050 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 281086 T + p^{6} T^{2} )^{2} \) |
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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
−11.70024162155993725173622569610, −11.19262266883399505293752683071, −10.69766134325806103040588097787, −10.48205384685107958544292833803, −10.04181287163406750227337818160, −9.081933227418080509057283708698, −8.640989358195149584938553670282, −8.317421652675071035950448798649, −7.55669228165606360592259965474, −6.65928994522426545702929126005, −6.43405612232792369341128446141, −6.04205749649615516870674866829, −5.53248234550219966178354264792, −4.71785903879708832661512800125, −4.27385367698987898980544536043, −3.61836347489956775401733642947, −2.80357064097396915466721767734, −1.42103927080781707987557928088, −1.38090979098309881164155549510, −0.27190927060308537420611369931,
0.27190927060308537420611369931, 1.38090979098309881164155549510, 1.42103927080781707987557928088, 2.80357064097396915466721767734, 3.61836347489956775401733642947, 4.27385367698987898980544536043, 4.71785903879708832661512800125, 5.53248234550219966178354264792, 6.04205749649615516870674866829, 6.43405612232792369341128446141, 6.65928994522426545702929126005, 7.55669228165606360592259965474, 8.317421652675071035950448798649, 8.640989358195149584938553670282, 9.081933227418080509057283708698, 10.04181287163406750227337818160, 10.48205384685107958544292833803, 10.69766134325806103040588097787, 11.19262266883399505293752683071, 11.70024162155993725173622569610
