L(s) = 1 | + 90·3-s + 3.50e3·7-s + 1.53e3·9-s − 5.14e4·13-s + 3.78e4·19-s + 3.15e5·21-s + 7.30e5·25-s − 4.51e5·27-s + 7.02e5·31-s − 2.67e6·37-s − 4.63e6·39-s − 7.05e6·43-s − 2.34e6·49-s + 3.40e6·57-s − 1.50e6·61-s + 5.38e6·63-s + 4.53e6·67-s + 5.53e7·73-s + 6.57e7·75-s + 4.59e7·79-s − 5.07e7·81-s − 1.80e8·91-s + 6.32e7·93-s + 2.94e8·97-s + 3.32e8·103-s + 2.19e8·109-s − 2.40e8·111-s + ⋯ |
L(s) = 1 | + 10/9·3-s + 1.45·7-s + 0.234·9-s − 1.80·13-s + 0.290·19-s + 1.61·21-s + 1.87·25-s − 0.850·27-s + 0.761·31-s − 1.42·37-s − 2.00·39-s − 2.06·43-s − 0.406·49-s + 0.322·57-s − 0.108·61-s + 0.341·63-s + 0.225·67-s + 1.94·73-s + 2.07·75-s + 1.18·79-s − 1.17·81-s − 2.62·91-s + 0.845·93-s + 3.32·97-s + 2.95·103-s + 1.55·109-s − 1.58·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(4.325197572\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.325197572\) |
\(L(5)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 10 p^{2} T + p^{8} T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 29234 p^{2} T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 250 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 380283362 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 25730 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8342551298 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 18938 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 64711613182 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 788066452322 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11338 p T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1335170 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12452468931842 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3526150 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30967680304898 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 80936075395298 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 105562517046242 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 753602 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2268890 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1001758688017922 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 27672770 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 22980982 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2352070843223138 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2600204109557762 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 147271010 T + p^{8} T^{2} )^{2} \) |
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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
−11.41374147273439395643806029111, −10.78027315923164073354451235389, −10.16087794975541997920810609144, −9.900250127812971928510953078256, −9.115432885666015484659206527134, −8.821862544453052927987017647861, −8.197969956219001378238944684681, −7.966808776946808401961197392347, −7.34540796917537938650444444446, −6.92565056130325057527748665359, −6.23492168679873949409838594243, −5.17020925726390311216297228226, −4.87415422025947824325835258399, −4.68283541085945721547247226635, −3.41550919490135299273803049760, −3.24743228928982274160706143898, −2.20009132927021261601765470668, −2.11468252608290202986388957272, −1.24576636162076006430012520972, −0.44415850645929573771848680659,
0.44415850645929573771848680659, 1.24576636162076006430012520972, 2.11468252608290202986388957272, 2.20009132927021261601765470668, 3.24743228928982274160706143898, 3.41550919490135299273803049760, 4.68283541085945721547247226635, 4.87415422025947824325835258399, 5.17020925726390311216297228226, 6.23492168679873949409838594243, 6.92565056130325057527748665359, 7.34540796917537938650444444446, 7.966808776946808401961197392347, 8.197969956219001378238944684681, 8.821862544453052927987017647861, 9.115432885666015484659206527134, 9.900250127812971928510953078256, 10.16087794975541997920810609144, 10.78027315923164073354451235389, 11.41374147273439395643806029111