| L(s) = 1 | + 104·4-s + 520·7-s + 1.37e4·13-s − 5.56e3·16-s + 6.63e4·19-s − 6.40e4·25-s + 5.40e4·28-s + 3.01e3·31-s − 7.61e5·37-s + 1.52e4·43-s − 1.44e6·49-s + 1.43e6·52-s − 1.97e6·61-s − 2.28e6·64-s + 7.71e6·67-s − 4.00e6·73-s + 6.90e6·76-s + 5.39e6·79-s + 7.16e6·91-s − 2.59e7·97-s − 6.66e6·100-s + 1.01e7·103-s + 1.31e7·109-s − 2.89e6·112-s − 2.11e6·121-s + 3.13e5·124-s + 127-s + ⋯ |
| L(s) = 1 | + 0.812·4-s + 0.573·7-s + 1.73·13-s − 0.339·16-s + 2.21·19-s − 0.820·25-s + 0.465·28-s + 0.0181·31-s − 2.47·37-s + 0.0293·43-s − 1.75·49-s + 1.41·52-s − 1.11·61-s − 1.08·64-s + 3.13·67-s − 1.20·73-s + 1.80·76-s + 1.23·79-s + 0.996·91-s − 2.88·97-s − 0.666·100-s + 0.914·103-s + 0.975·109-s − 0.194·112-s − 0.108·121-s + 0.0147·124-s + 1.27·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.148091097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148091097\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 13 p^{3} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 12818 p T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 260 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2110342 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 530 p T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 259975906 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 33176 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5812848334 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15420328618 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 1508 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 380770 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 381757891762 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7640 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 693036689566 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1288434979834 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2355536454362 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 988858 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3857360 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 332728892782 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2004730 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2699684 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 46919519671414 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 28535629791058 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12957490 T + p^{7} T^{2} )^{2} \) |
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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
−20.38969099516560214293303929997, −19.54052804846231077845583977325, −18.62599180950191572881612873522, −18.01101674480815129753340020094, −17.41608347834365136523208186635, −16.09721304205976252787375380795, −15.99439371957814388436451179729, −15.22933754465762612426156489372, −13.90537092037521732118852078884, −13.72863634433878072038780045570, −12.34634256857787230273205608094, −11.42215001987623811385701429810, −11.08169161750989189708347039487, −9.854453736333627561557481571909, −8.684430906186519251513441244538, −7.64730526586133418050975846402, −6.55569468139126945446386505796, −5.31877812668296384837894467469, −3.45528160667815042061928596501, −1.53590339893828668060725860717,
1.53590339893828668060725860717, 3.45528160667815042061928596501, 5.31877812668296384837894467469, 6.55569468139126945446386505796, 7.64730526586133418050975846402, 8.684430906186519251513441244538, 9.854453736333627561557481571909, 11.08169161750989189708347039487, 11.42215001987623811385701429810, 12.34634256857787230273205608094, 13.72863634433878072038780045570, 13.90537092037521732118852078884, 15.22933754465762612426156489372, 15.99439371957814388436451179729, 16.09721304205976252787375380795, 17.41608347834365136523208186635, 18.01101674480815129753340020094, 18.62599180950191572881612873522, 19.54052804846231077845583977325, 20.38969099516560214293303929997
