| L(s) = 1 | − 4·2-s + 3-s + 10·4-s − 5-s − 4·6-s − 20·8-s − 2·9-s + 4·10-s − 11-s + 10·12-s − 9·13-s − 15-s + 35·16-s + 8·18-s − 2·19-s − 10·20-s + 4·22-s − 4·23-s − 20·24-s − 10·25-s + 36·26-s + 3·27-s − 4·29-s + 4·30-s − 33·31-s − 56·32-s − 33-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 5·4-s − 0.447·5-s − 1.63·6-s − 7.07·8-s − 2/3·9-s + 1.26·10-s − 0.301·11-s + 2.88·12-s − 2.49·13-s − 0.258·15-s + 35/4·16-s + 1.88·18-s − 0.458·19-s − 2.23·20-s + 0.852·22-s − 0.834·23-s − 4.08·24-s − 2·25-s + 7.06·26-s + 0.577·27-s − 0.742·29-s + 0.730·30-s − 5.92·31-s − 9.89·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + p T^{2} - 8 T^{3} + 4 p T^{4} - 8 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + T + 11 T^{2} + 14 T^{3} + 72 T^{4} + 14 p T^{5} + 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 8 T^{2} - 10 T^{3} + 22 T^{4} - 10 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + T + 37 T^{2} + 28 T^{3} + 580 T^{4} + 28 p T^{5} + 37 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 9 T + 75 T^{2} + 362 T^{3} + 1606 T^{4} + 362 p T^{5} + 75 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 38 T^{2} - 82 T^{3} + 658 T^{4} - 82 p T^{5} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 T + 60 T^{2} + 86 T^{3} + 1590 T^{4} + 86 p T^{5} + 60 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 33 T + 525 T^{2} + 5192 T^{3} + 34756 T^{4} + 5192 p T^{5} + 525 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 12 T + 136 T^{2} + 850 T^{3} + 6274 T^{4} + 850 p T^{5} + 136 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 6 T + 92 T^{2} + 388 T^{3} + 4978 T^{4} + 388 p T^{5} + 92 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 5 T + 105 T^{2} + 464 T^{3} + 6540 T^{4} + 464 p T^{5} + 105 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 3 T + 77 T^{2} - 580 T^{3} + 2788 T^{4} - 580 p T^{5} + 77 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 9 T + 59 T^{2} + 166 T^{3} + 88 T^{4} + 166 p T^{5} + 59 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 2 T + 208 T^{2} - 314 T^{3} + 17710 T^{4} - 314 p T^{5} + 208 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 24 T + 440 T^{2} + 5006 T^{3} + 46762 T^{4} + 5006 p T^{5} + 440 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 8 T + 262 T^{2} + 1602 T^{3} + 26134 T^{4} + 1602 p T^{5} + 262 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 6 T + 184 T^{2} - 1414 T^{3} + 16494 T^{4} - 1414 p T^{5} + 184 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 22 T + 388 T^{2} + 4436 T^{3} + 44874 T^{4} + 4436 p T^{5} + 388 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 29 T + 549 T^{2} + 7308 T^{3} + 74166 T^{4} + 7308 p T^{5} + 549 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 12 T + 316 T^{2} - 2588 T^{3} + 38534 T^{4} - 2588 p T^{5} + 316 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 20 T + 338 T^{2} + 3298 T^{3} + 36258 T^{4} + 3298 p T^{5} + 338 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 4 T + 244 T^{2} - 1100 T^{3} + 31590 T^{4} - 1100 p T^{5} + 244 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.54655838570794790981591910358, −7.37612943913926103959057289748, −7.17668743267546553772728346750, −6.81578103487490224136151927463, −6.60121496703643990231141691070, −6.45140438903936622485977240374, −6.06144937696711355189013464842, −5.84201936963953773534347235372, −5.64815293325255695566949608505, −5.32021733490648059013101135098, −5.29869615302320818252843018020, −5.11874407967946673073661095953, −4.46342232857400571110385713159, −4.44754608053833236539457620957, −4.01424333125866348756512151561, −3.63403408190170291015972460662, −3.50493612867505307537071246649, −3.20071139765482765545602136663, −2.95558543977636762496948119907, −2.73545943910343702383571569010, −2.36185942027396913971639140173, −1.94930452623511151406029997065, −1.77274594105414892106599143038, −1.66745421051693898853852100927, −1.51934110215646368009139690093, 0, 0, 0, 0,
1.51934110215646368009139690093, 1.66745421051693898853852100927, 1.77274594105414892106599143038, 1.94930452623511151406029997065, 2.36185942027396913971639140173, 2.73545943910343702383571569010, 2.95558543977636762496948119907, 3.20071139765482765545602136663, 3.50493612867505307537071246649, 3.63403408190170291015972460662, 4.01424333125866348756512151561, 4.44754608053833236539457620957, 4.46342232857400571110385713159, 5.11874407967946673073661095953, 5.29869615302320818252843018020, 5.32021733490648059013101135098, 5.64815293325255695566949608505, 5.84201936963953773534347235372, 6.06144937696711355189013464842, 6.45140438903936622485977240374, 6.60121496703643990231141691070, 6.81578103487490224136151927463, 7.17668743267546553772728346750, 7.37612943913926103959057289748, 7.54655838570794790981591910358
