| L(s) = 1 | − 2.44e3·7-s + 2.43e3·9-s − 3.25e4·17-s + 3.15e3·23-s − 2.86e4·25-s − 5.02e5·31-s + 6.38e5·41-s − 5.68e5·47-s + 2.84e6·49-s − 5.96e6·63-s − 5.42e6·71-s + 1.13e7·73-s + 1.02e7·79-s + 1.16e6·81-s − 2.32e7·89-s + 2.18e7·97-s + 1.75e7·103-s + 7.81e6·113-s + 7.96e7·119-s + 2.89e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 7.93e7·153-s + ⋯ |
| L(s) = 1 | − 2.69·7-s + 1.11·9-s − 1.60·17-s + 0.0540·23-s − 0.366·25-s − 3.03·31-s + 1.44·41-s − 0.798·47-s + 3.45·49-s − 3.00·63-s − 1.79·71-s + 3.41·73-s + 2.33·79-s + 0.242·81-s − 3.49·89-s + 2.43·97-s + 1.57·103-s + 0.509·113-s + 4.33·119-s + 1.48·121-s − 1.79·153-s − 0.145·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.03927008148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03927008148\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2438 T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 1146 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 1224 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 28963446 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 88067110 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 16270 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 1757756902 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 1576 T + p^{7} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 19410578374 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 251360 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 187124488022 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 319398 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 38417641270 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 284112 T + p^{7} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2261769571830 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4171710557334 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5502596935942 T^{2} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9591306324218 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2710792 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5670854 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5124176 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 51827394614118 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 11605674 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10931618 T + p^{7} T^{2} )^{2} \) |
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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
−11.05427787668024872803435300870, −10.43450434066347640124815664741, −9.942233750451114837072807622291, −9.557774207502462719246051101584, −9.106110723318675247021194983352, −9.035655730246680611652254980065, −8.008050335346126832238781656264, −7.39160426487094807734956181841, −6.83399243876306077781204902739, −6.76291367505245050973876733006, −6.06795520736222489639144386032, −5.67958363933888694862140998554, −4.78591589548846348652510476912, −4.14884121723716263177511383389, −3.56379119241209181949840104877, −3.32643230903452992895782097850, −2.35770855143212037865308355223, −1.95223506056867638052484067988, −0.908462150807962661940769332009, −0.05404320869199923687138726453,
0.05404320869199923687138726453, 0.908462150807962661940769332009, 1.95223506056867638052484067988, 2.35770855143212037865308355223, 3.32643230903452992895782097850, 3.56379119241209181949840104877, 4.14884121723716263177511383389, 4.78591589548846348652510476912, 5.67958363933888694862140998554, 6.06795520736222489639144386032, 6.76291367505245050973876733006, 6.83399243876306077781204902739, 7.39160426487094807734956181841, 8.008050335346126832238781656264, 9.035655730246680611652254980065, 9.106110723318675247021194983352, 9.557774207502462719246051101584, 9.942233750451114837072807622291, 10.43450434066347640124815664741, 11.05427787668024872803435300870
