| L(s) = 1 | − 24·5-s − 44·11-s − 26·13-s − 164·17-s − 48·19-s + 8·23-s + 238·25-s − 404·29-s − 40·31-s − 100·37-s − 200·41-s + 616·43-s − 324·47-s − 630·49-s + 164·53-s + 1.05e3·55-s + 140·59-s + 628·61-s + 624·65-s + 472·67-s + 428·71-s − 900·73-s + 432·79-s − 1.38e3·83-s + 3.93e3·85-s − 960·89-s + 1.15e3·95-s + ⋯ |
| L(s) = 1 | − 2.14·5-s − 1.20·11-s − 0.554·13-s − 2.33·17-s − 0.579·19-s + 0.0725·23-s + 1.90·25-s − 2.58·29-s − 0.231·31-s − 0.444·37-s − 0.761·41-s + 2.18·43-s − 1.00·47-s − 1.83·49-s + 0.425·53-s + 2.58·55-s + 0.308·59-s + 1.31·61-s + 1.19·65-s + 0.860·67-s + 0.715·71-s − 1.44·73-s + 0.615·79-s − 1.83·83-s + 5.02·85-s − 1.14·89-s + 1.24·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3921135210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3921135210\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 24 T + 338 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 90 p T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 p T + 1130 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 164 T + 16326 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 48 T + 14238 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T - 7906 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 404 T + 81518 T^{2} + 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T + 50518 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 100 T + 92830 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 200 T + 24138 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 616 T + 216022 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 324 T + 219554 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 164 T + 102878 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 140 T + 393258 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 628 T + 293614 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 472 T + 252622 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 428 T + 662834 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 900 T + 899670 T^{2} + 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 432 T + 924318 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1388 T + 1567866 T^{2} + 1388 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 960 T + 1134938 T^{2} + 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 532 T + 1218502 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.876945323159182188990640394562, −8.650898837601987777580682769464, −8.167573743537750792128700529135, −7.931310220444142018020282359267, −7.44608868798587376102245828012, −7.33146216276253504224145221228, −6.66558676291265176324968653634, −6.64073243704483471947367022117, −5.72633289221930349351870121153, −5.39645402726394208044202571783, −4.93704712677491771613834061245, −4.41043603743051835093402491814, −4.01201843596050934159745753787, −3.95399500646417534315877385824, −3.16208163448787652993799768728, −2.79126862585477926126022175238, −2.03029112079314287833769872889, −1.84417856886970283681361388643, −0.45637229724694903303123356538, −0.27290699877579704263041683000,
0.27290699877579704263041683000, 0.45637229724694903303123356538, 1.84417856886970283681361388643, 2.03029112079314287833769872889, 2.79126862585477926126022175238, 3.16208163448787652993799768728, 3.95399500646417534315877385824, 4.01201843596050934159745753787, 4.41043603743051835093402491814, 4.93704712677491771613834061245, 5.39645402726394208044202571783, 5.72633289221930349351870121153, 6.64073243704483471947367022117, 6.66558676291265176324968653634, 7.33146216276253504224145221228, 7.44608868798587376102245828012, 7.931310220444142018020282359267, 8.167573743537750792128700529135, 8.650898837601987777580682769464, 8.876945323159182188990640394562
