| L(s) = 1 | + 8·2-s + 29·3-s + 48·4-s − 13·5-s + 232·6-s − 14·7-s + 256·8-s + 343·9-s − 104·10-s − 242·11-s + 1.39e3·12-s − 646·13-s − 112·14-s − 377·15-s + 1.28e3·16-s − 208·17-s + 2.74e3·18-s − 2.14e3·19-s − 624·20-s − 406·21-s − 1.93e3·22-s + 349·23-s + 7.42e3·24-s − 1.16e3·25-s − 5.16e3·26-s + 2.55e3·27-s − 672·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.86·3-s + 3/2·4-s − 0.232·5-s + 2.63·6-s − 0.107·7-s + 1.41·8-s + 1.41·9-s − 0.328·10-s − 0.603·11-s + 2.79·12-s − 1.06·13-s − 0.152·14-s − 0.432·15-s + 5/4·16-s − 0.174·17-s + 1.99·18-s − 1.36·19-s − 0.348·20-s − 0.200·21-s − 0.852·22-s + 0.137·23-s + 2.63·24-s − 0.373·25-s − 1.49·26-s + 0.673·27-s − 0.161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.205598830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.205598830\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 29 T + 166 p T^{2} - 29 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 13 T + 1336 T^{2} + 13 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 p T + 26526 T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 646 T + 827090 T^{2} + 646 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 208 T - 197762 T^{2} + 208 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2148 T + 5721862 T^{2} + 2148 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 349 T + 12893422 T^{2} - 349 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4422 T + 19354042 T^{2} - 4422 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14381 T + 3464234 p T^{2} - 14381 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4267 T + 138838388 T^{2} + 4267 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10110 T + 53425570 T^{2} + 10110 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9798 T + 274223662 T^{2} + 9798 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21144 T + 551158750 T^{2} - 21144 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 39584 T + 1198268902 T^{2} - 39584 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 90951 T + 3475885954 T^{2} - 90951 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 29550 T + 1852642210 T^{2} + 29550 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 64149 T + 3285930058 T^{2} + 64149 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 23583 T + 295186750 T^{2} - 23583 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 39058 T + 2830212410 T^{2} + 39058 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 54974 T + 4805310654 T^{2} + 54974 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 29986 T + 2580164782 T^{2} - 29986 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18047 T + 124167368 p T^{2} + 18047 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 30309 T + 4031708980 T^{2} + 30309 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−17.12438372213077980661329149263, −16.30706976744065583126960960649, −15.43056212579708272017089159751, −15.25794832559711428320081936732, −14.59242730418240770057700009643, −14.15743857610710914451927738468, −13.39314689328576622679316898533, −13.20581141753512836387993607681, −12.15979829818438029257858199458, −11.76608608727053001701397026964, −10.40098556533880422791278728158, −10.01665212527041940050614168686, −8.591638839997830687675856108655, −8.296519315262559466806630929154, −7.27773672779741335234686405176, −6.42905153460543352867498148551, −4.98870416990803796511826858960, −4.10688611555096671578322516381, −2.90752767767707737202459203580, −2.37687753792476808710949628146,
2.37687753792476808710949628146, 2.90752767767707737202459203580, 4.10688611555096671578322516381, 4.98870416990803796511826858960, 6.42905153460543352867498148551, 7.27773672779741335234686405176, 8.296519315262559466806630929154, 8.591638839997830687675856108655, 10.01665212527041940050614168686, 10.40098556533880422791278728158, 11.76608608727053001701397026964, 12.15979829818438029257858199458, 13.20581141753512836387993607681, 13.39314689328576622679316898533, 14.15743857610710914451927738468, 14.59242730418240770057700009643, 15.25794832559711428320081936732, 15.43056212579708272017089159751, 16.30706976744065583126960960649, 17.12438372213077980661329149263
