| L(s) = 1 | + 3·2-s + 4·4-s + 4·7-s + 3·8-s − 9-s + 2·11-s − 4·13-s + 12·14-s + 3·16-s + 3·17-s − 3·18-s + 6·22-s − 2·23-s − 12·26-s + 16·28-s − 12·29-s − 3·31-s + 6·32-s + 9·34-s − 4·36-s + 15·37-s + 15·41-s − 43-s + 8·44-s − 6·46-s + 13·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 1.51·7-s + 1.06·8-s − 1/3·9-s + 0.603·11-s − 1.10·13-s + 3.20·14-s + 3/4·16-s + 0.727·17-s − 0.707·18-s + 1.27·22-s − 0.417·23-s − 2.35·26-s + 3.02·28-s − 2.22·29-s − 0.538·31-s + 1.06·32-s + 1.54·34-s − 2/3·36-s + 2.46·37-s + 2.34·41-s − 0.152·43-s + 1.20·44-s − 0.884·46-s + 1.89·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.36941737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.36941737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 15 T + 119 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 137 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 13 T + 135 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 161 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 203 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 155 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 17 T + 265 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−7.70946533386677580161801671705, −7.53199758408307633699694527820, −7.42544825868587483579585807805, −6.65800630512978560427129183879, −6.23808047807277525242572979983, −6.15172373135778251105009087537, −5.56275845217199680373185068814, −5.36523198740411731674930401879, −5.15533855330473720133972005760, −4.86032696566018611235211145461, −4.33576223688831134920011937676, −4.03933060691991415349575302132, −3.91745396200519847767147674306, −3.57101228384936875797749352760, −2.85071817820189945034282567084, −2.58267636702775333577416494771, −2.18123886133608093418840986860, −1.73361146595770377618383141504, −1.13781103842675887960957823458, −0.58815393720946770600808150623,
0.58815393720946770600808150623, 1.13781103842675887960957823458, 1.73361146595770377618383141504, 2.18123886133608093418840986860, 2.58267636702775333577416494771, 2.85071817820189945034282567084, 3.57101228384936875797749352760, 3.91745396200519847767147674306, 4.03933060691991415349575302132, 4.33576223688831134920011937676, 4.86032696566018611235211145461, 5.15533855330473720133972005760, 5.36523198740411731674930401879, 5.56275845217199680373185068814, 6.15172373135778251105009087537, 6.23808047807277525242572979983, 6.65800630512978560427129183879, 7.42544825868587483579585807805, 7.53199758408307633699694527820, 7.70946533386677580161801671705
