| L(s) = 1 | + 16·2-s − 23·3-s + 192·4-s + 331·5-s − 368·6-s + 1.79e3·7-s + 2.04e3·8-s − 257·9-s + 5.29e3·10-s + 2.66e3·11-s − 4.41e3·12-s − 5.40e3·13-s + 2.87e4·14-s − 7.61e3·15-s + 2.04e4·16-s + 1.50e4·17-s − 4.11e3·18-s + 1.69e4·19-s + 6.35e4·20-s − 4.12e4·21-s + 4.25e4·22-s − 5.13e4·23-s − 4.71e4·24-s − 7.03e4·25-s − 8.64e4·26-s − 2.63e4·27-s + 3.44e5·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.491·3-s + 3/2·4-s + 1.18·5-s − 0.695·6-s + 1.97·7-s + 1.41·8-s − 0.117·9-s + 1.67·10-s + 0.603·11-s − 0.737·12-s − 0.682·13-s + 2.79·14-s − 0.582·15-s + 5/4·16-s + 0.742·17-s − 0.166·18-s + 0.565·19-s + 1.77·20-s − 0.972·21-s + 0.852·22-s − 0.880·23-s − 0.695·24-s − 0.900·25-s − 0.965·26-s − 0.257·27-s + 2.96·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.726908037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.726908037\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 23 T + 262 p T^{2} + 23 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 331 T + 35984 p T^{2} - 331 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 1794 T + 1722526 T^{2} - 1794 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5406 T + 81002482 T^{2} + 5406 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 15032 T + 669964558 T^{2} - 15032 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 16916 T + 987790358 T^{2} - 16916 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 51351 T + 6272862742 T^{2} + 51351 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 207130 T + 43242940618 T^{2} + 207130 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 19071 T + 48877201726 T^{2} + 19071 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 351333 T + 178095011932 T^{2} - 351333 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 123610 T + 154875127858 T^{2} - 123610 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 159822 T - 48316439426 T^{2} + 159822 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 451160 T + 940360514590 T^{2} - 451160 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1260832 T + 2424528378934 T^{2} + 1260832 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 887547 T + 4412738990098 T^{2} - 887547 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 597918 T + 5828558606962 T^{2} + 597918 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2864711 T + 9630590949434 T^{2} - 2864711 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1306267 T + 12435173811382 T^{2} - 1306267 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4577530 T + 24133774153130 T^{2} + 4577530 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2946342 T + 29218466187838 T^{2} + 2946342 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9965450 T + 76820028095230 T^{2} - 9965450 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10185377 T + 97278081607768 T^{2} - 10185377 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 27765477 T + 354321137728852 T^{2} + 27765477 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−16.78140729005711663746400424471, −16.06702743645330790232420830992, −14.95222047498426488598763333328, −14.77541328287213662477797279434, −13.94911154603344225085101670922, −13.88789065474692011749096096171, −12.88686609847004463585115620556, −12.09253308708077720115420804368, −11.47286221488653987237158568944, −11.20546971316346806924742106620, −10.11131630818385781309617493968, −9.430533006243205321288840358069, −7.970939837231862943446533645293, −7.43542258591045502889278444378, −6.06354102838105549600893493993, −5.59929163749389598532632775262, −4.91297575058219524071789527774, −3.90971896388252046703370907284, −2.19640397551480146584135652672, −1.51038172733195659926201817794,
1.51038172733195659926201817794, 2.19640397551480146584135652672, 3.90971896388252046703370907284, 4.91297575058219524071789527774, 5.59929163749389598532632775262, 6.06354102838105549600893493993, 7.43542258591045502889278444378, 7.970939837231862943446533645293, 9.430533006243205321288840358069, 10.11131630818385781309617493968, 11.20546971316346806924742106620, 11.47286221488653987237158568944, 12.09253308708077720115420804368, 12.88686609847004463585115620556, 13.88789065474692011749096096171, 13.94911154603344225085101670922, 14.77541328287213662477797279434, 14.95222047498426488598763333328, 16.06702743645330790232420830992, 16.78140729005711663746400424471
