| L(s) = 1 | − 2.18e3·9-s − 2.92e4·13-s + 1.56e5·25-s − 5.59e5·37-s − 1.38e6·49-s + 7.07e6·61-s + 1.25e7·73-s + 4.78e6·81-s − 2.44e7·97-s − 3.36e7·109-s + 6.39e7·117-s − 3.89e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.15e8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s − 3.68·13-s + 2·25-s − 1.81·37-s − 1.68·49-s + 3.98·61-s + 3.77·73-s + 81-s − 2.72·97-s − 2.48·109-s + 3.68·117-s − 2·121-s + 8.21·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.09217770622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09217770622\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 508 T + p^{7} T^{2} )( 1 + 508 T + p^{7} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 14614 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 57448 T + p^{7} T^{2} )( 1 + 57448 T + p^{7} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 178916 T + p^{7} T^{2} )( 1 + 178916 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 279710 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 1035224 T + p^{7} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3535546 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 385072 T + p^{7} T^{2} )( 1 + 385072 T + p^{7} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6274810 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8763044 T + p^{7} T^{2} )( 1 + 8763044 T + p^{7} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12245198 T + p^{7} T^{2} )^{2} \) |
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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
−12.06307291823413843129222722742, −10.98533914181917178306603362293, −10.52472014436010128017524532382, −9.830200720263015215291814085503, −9.659264011391398292781591133079, −9.117614824531049181911078640840, −8.331233678408138834740670717036, −8.101478860613731474127700911633, −7.31895045093175899519348238932, −6.73375637921156299919770392406, −6.73233544760550273040317385953, −5.31186854528034391930689751108, −5.16904429864291205140409461974, −4.91067308666795361561061468239, −3.88789405713229226942239345080, −3.09753818156138453747627976441, −2.41399973116681595484826347764, −2.28135334293542208660316386341, −1.02502713302394980122508437558, −0.083591046351202198967947105267,
0.083591046351202198967947105267, 1.02502713302394980122508437558, 2.28135334293542208660316386341, 2.41399973116681595484826347764, 3.09753818156138453747627976441, 3.88789405713229226942239345080, 4.91067308666795361561061468239, 5.16904429864291205140409461974, 5.31186854528034391930689751108, 6.73233544760550273040317385953, 6.73375637921156299919770392406, 7.31895045093175899519348238932, 8.101478860613731474127700911633, 8.331233678408138834740670717036, 9.117614824531049181911078640840, 9.659264011391398292781591133079, 9.830200720263015215291814085503, 10.52472014436010128017524532382, 10.98533914181917178306603362293, 12.06307291823413843129222722742
