| L(s) = 1 | − 1.02e3·5-s − 7.03e3·9-s − 5.54e4·13-s + 1.00e5·17-s − 950·25-s + 1.09e5·29-s + 1.58e6·37-s − 1.51e5·41-s + 7.17e6·45-s + 4.99e6·49-s + 2.23e7·53-s − 4.76e7·61-s + 5.65e7·65-s + 1.30e7·73-s + 6.48e6·81-s − 1.02e8·85-s + 1.73e8·89-s − 9.33e7·97-s + 1.31e8·101-s − 3.12e8·109-s + 4.72e8·113-s + 3.90e8·117-s + 6.13e7·121-s + 6.64e8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.63·5-s − 1.07·9-s − 1.94·13-s + 1.20·17-s − 0.00243·25-s + 0.155·29-s + 0.847·37-s − 0.0534·41-s + 1.75·45-s + 0.866·49-s + 2.83·53-s − 3.44·61-s + 3.16·65-s + 0.458·73-s + 0.150·81-s − 1.96·85-s + 2.76·89-s − 1.05·97-s + 1.26·101-s − 2.21·109-s + 2.89·113-s + 2.08·117-s + 0.285·121-s + 2.72·125-s − 0.253·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6408731803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6408731803\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 26 p T + p^{8} T^{2} )( 1 + 26 p T + p^{8} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 102 p T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 713966 p T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 61301762 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 27710 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 50370 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22168990082 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 237285218 p^{2} T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 54978 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 323644730882 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 793730 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 75582 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 23126751997442 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 39399744456962 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 11166210 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 182995617327358 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 23826622 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 755964429617282 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1189851873275522 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6516610 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 653831579109122 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 890071220357758 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 86795778 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 46670270 T + p^{8} T^{2} )^{2} \) |
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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
−17.38624172233306483745881687732, −16.76646688473833333705190776647, −16.40495392749809605101946713050, −15.30052339723711556560907967474, −15.04648351514145639173738149473, −14.35818415366575748952044509475, −13.58529294497711327043524366998, −12.27801924036190834159341145690, −12.05418020255978570190492316653, −11.53329504294888990292621947213, −10.51838123367464733612617225198, −9.649148753062120836408863498092, −8.645736225117550084400133122170, −7.62006934548952151741623958571, −7.49769515905505937595628963416, −5.88899677477203516236891189083, −4.81358410003335607058665818201, −3.70734191623645052316818471976, −2.60379807457954539457520050565, −0.43933865474779621515255564421,
0.43933865474779621515255564421, 2.60379807457954539457520050565, 3.70734191623645052316818471976, 4.81358410003335607058665818201, 5.88899677477203516236891189083, 7.49769515905505937595628963416, 7.62006934548952151741623958571, 8.645736225117550084400133122170, 9.649148753062120836408863498092, 10.51838123367464733612617225198, 11.53329504294888990292621947213, 12.05418020255978570190492316653, 12.27801924036190834159341145690, 13.58529294497711327043524366998, 14.35818415366575748952044509475, 15.04648351514145639173738149473, 15.30052339723711556560907967474, 16.40495392749809605101946713050, 16.76646688473833333705190776647, 17.38624172233306483745881687732
