| L(s) = 1 | + 1.25e3·5-s + 928·7-s + 2.62e4·11-s − 2.81e4·13-s + 5.78e5·17-s − 6.37e5·19-s + 6.53e5·23-s + 1.17e6·25-s + 4.87e6·29-s + 7.66e5·31-s + 1.16e6·35-s − 2.06e7·37-s + 5.39e7·41-s − 3.10e7·43-s + 4.67e7·47-s − 7.84e7·49-s + 1.41e7·53-s + 3.27e7·55-s + 2.95e7·59-s − 1.39e8·61-s − 3.51e7·65-s + 2.67e7·67-s + 8.86e7·71-s + 5.78e8·73-s + 2.43e7·77-s + 5.35e8·79-s − 3.16e8·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.146·7-s + 0.539·11-s − 0.272·13-s + 1.67·17-s − 1.12·19-s + 0.486·23-s + 3/5·25-s + 1.27·29-s + 0.149·31-s + 0.130·35-s − 1.80·37-s + 2.97·41-s − 1.38·43-s + 1.39·47-s − 1.94·49-s + 0.246·53-s + 0.482·55-s + 0.317·59-s − 1.29·61-s − 0.244·65-s + 0.162·67-s + 0.414·71-s + 2.38·73-s + 0.0788·77-s + 1.54·79-s − 0.732·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.682200509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.682200509\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 928 T + 11325330 p T^{2} - 928 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 26208 T + 438989398 T^{2} - 26208 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 28100 T - 10380507954 T^{2} + 28100 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 578172 T + 316125101590 T^{2} - 578172 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 637280 T + 317310575958 T^{2} + 637280 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 653352 T + 1765105213102 T^{2} - 653352 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4872012 T + 31147347103774 T^{2} - 4872012 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 766408 T + 50111517469758 T^{2} - 766408 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 20625476 T + 304595639215998 T^{2} + 20625476 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 53910780 T + 1374630544963222 T^{2} - 53910780 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 31095752 T + 555256532861862 T^{2} + 31095752 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 46765848 T + 2668743212860510 T^{2} - 46765848 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14170044 T - 1303864544335250 T^{2} - 14170044 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 29507232 T + 10336284191236534 T^{2} - 29507232 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 139916996 T + 25813132921791486 T^{2} + 139916996 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 26779192 T + 36053829938132310 T^{2} - 26779192 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 88690560 T + 61894888086856462 T^{2} - 88690560 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 578530804 T + 195628023410640630 T^{2} - 578530804 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 535121176 T + 272000696108593182 T^{2} - 535121176 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 316673688 T + 299480779966372342 T^{2} + 316673688 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 320304660 T + 268195935984914518 T^{2} + 320304660 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2182180996 T + 2708056121570461638 T^{2} - 2182180996 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−10.99244106820272357808134605649, −10.81162225475002051307274009836, −10.00362873315657665237961175950, −9.982694890427131558836165657799, −9.161226741087567546519980252980, −8.894826497489603680462906872142, −8.163694397109723335740894642942, −7.76615666624817520676981673082, −7.03284739645898168494601990705, −6.53320198217128737599962793497, −6.04330041417453270715377597847, −5.52079620920177338400461966355, −4.86064747551046333992329150525, −4.44616502225836755690282835142, −3.48124927373072840396562588828, −3.12382648347280768229853699959, −2.20082492210796827964552941823, −1.83665175006970620680657373768, −0.952569127666371461038339396450, −0.63628638524191576551791276821,
0.63628638524191576551791276821, 0.952569127666371461038339396450, 1.83665175006970620680657373768, 2.20082492210796827964552941823, 3.12382648347280768229853699959, 3.48124927373072840396562588828, 4.44616502225836755690282835142, 4.86064747551046333992329150525, 5.52079620920177338400461966355, 6.04330041417453270715377597847, 6.53320198217128737599962793497, 7.03284739645898168494601990705, 7.76615666624817520676981673082, 8.163694397109723335740894642942, 8.894826497489603680462906872142, 9.161226741087567546519980252980, 9.982694890427131558836165657799, 10.00362873315657665237961175950, 10.81162225475002051307274009836, 10.99244106820272357808134605649
