| L(s) = 1 | − 8·2-s + 48·4-s + 12·5-s − 176·7-s − 256·8-s − 96·10-s + 540·11-s − 446·13-s + 1.40e3·14-s + 1.28e3·16-s + 1.56e3·17-s − 1.59e3·19-s + 576·20-s − 4.32e3·22-s + 1.40e3·23-s − 5.46e3·25-s + 3.56e3·26-s − 8.44e3·28-s + 2.34e3·29-s − 776·31-s − 6.14e3·32-s − 1.24e4·34-s − 2.11e3·35-s + 3.52e3·37-s + 1.27e4·38-s − 3.07e3·40-s + 2.78e4·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.214·5-s − 1.35·7-s − 1.41·8-s − 0.303·10-s + 1.34·11-s − 0.731·13-s + 1.91·14-s + 5/4·16-s + 1.30·17-s − 1.01·19-s + 0.321·20-s − 1.90·22-s + 0.553·23-s − 1.74·25-s + 1.03·26-s − 2.03·28-s + 0.516·29-s − 0.145·31-s − 1.06·32-s − 1.85·34-s − 0.291·35-s + 0.423·37-s + 1.43·38-s − 0.303·40-s + 2.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.355940441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.355940441\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 12 T + 5611 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 176 T + 28290 T^{2} + 176 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 540 T + 310330 T^{2} - 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 446 T + 60939 p T^{2} + 446 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1560 T + 3172687 T^{2} - 1560 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1592 T + 5183946 T^{2} + 1592 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1404 T + 515726 p T^{2} - 1404 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2340 T + 11651995 T^{2} - 2340 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 776 T - 33605346 T^{2} + 776 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3526 T + 28328583 T^{2} - 3526 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 27840 T + 421901410 T^{2} - 27840 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 23104 T + 371010858 T^{2} - 23104 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 38784 T + 16628018 p T^{2} - 38784 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 45672 T + 1133302570 T^{2} - 45672 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2160 T + 484762198 T^{2} - 2160 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8974 T + 1242523743 T^{2} - 8974 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 71656 T + 2902035498 T^{2} - 71656 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 348 p T + 3665651218 T^{2} - 348 p^{6} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 60322 T + 3938822259 T^{2} - 60322 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 74392 T + 6604781346 T^{2} - 74392 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 29496 T + 4384376038 T^{2} - 29496 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 110976 T + 7596045367 T^{2} + 110976 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5564 T + 16765832070 T^{2} + 5564 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.03841207444431840987372735741, −11.80993117723898863837476008524, −10.87946448795417220554065181197, −10.70585741094600976066697762862, −9.861308206657574285095991796343, −9.710370047099251418617261327368, −9.109843161912834324267414552487, −9.050050766718077960158925417302, −7.87790465447330709347748544127, −7.79397428472205200509569884867, −6.81364465655086795222324497437, −6.72775382521335988061329217131, −5.74638919873629288319345394592, −5.70719048738394998779657559128, −4.03650529814829765100049138428, −3.84033777305367365811212062605, −2.60832319291071103882930557286, −2.29499253719226004491340344218, −0.984217731142609464701635379009, −0.61729310905426951384852704153,
0.61729310905426951384852704153, 0.984217731142609464701635379009, 2.29499253719226004491340344218, 2.60832319291071103882930557286, 3.84033777305367365811212062605, 4.03650529814829765100049138428, 5.70719048738394998779657559128, 5.74638919873629288319345394592, 6.72775382521335988061329217131, 6.81364465655086795222324497437, 7.79397428472205200509569884867, 7.87790465447330709347748544127, 9.050050766718077960158925417302, 9.109843161912834324267414552487, 9.710370047099251418617261327368, 9.861308206657574285095991796343, 10.70585741094600976066697762862, 10.87946448795417220554065181197, 11.80993117723898863837476008524, 12.03841207444431840987372735741
