| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s + 6·7-s − 4·8-s − 9-s + 6·12-s + 6·13-s − 12·14-s + 5·16-s + 2·17-s + 2·18-s − 2·19-s + 12·21-s + 10·23-s − 8·24-s − 12·26-s − 6·27-s + 18·28-s − 6·29-s + 4·31-s − 6·32-s − 4·34-s − 3·36-s + 4·38-s + 12·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 2.26·7-s − 1.41·8-s − 1/3·9-s + 1.73·12-s + 1.66·13-s − 3.20·14-s + 5/4·16-s + 0.485·17-s + 0.471·18-s − 0.458·19-s + 2.61·21-s + 2.08·23-s − 1.63·24-s − 2.35·26-s − 1.15·27-s + 3.40·28-s − 1.11·29-s + 0.718·31-s − 1.06·32-s − 0.685·34-s − 1/2·36-s + 0.648·38-s + 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.530082599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.530082599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 24 T + 4 p T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.28077010141182019098845748019, −9.485047597210104391000956809384, −9.314120936575675239669110822531, −8.831862100813789181510536703591, −8.458968035015643325998359679496, −8.413752961397944145466971390681, −7.85092908730108193919178449370, −7.74352952259069312063347770358, −7.08470267100953588111221212276, −6.69205257398238111475743537251, −5.86080638297475991552836630505, −5.70804418276667345505510457016, −4.97743752742474768646814718854, −4.54338866634520780302783749167, −3.63150975539577213875955977149, −3.40785226109579241709062727415, −2.48002232869173389252567368619, −2.24874253997621326318976801540, −1.34184171380425132232223414114, −1.04897719567186889567602968690,
1.04897719567186889567602968690, 1.34184171380425132232223414114, 2.24874253997621326318976801540, 2.48002232869173389252567368619, 3.40785226109579241709062727415, 3.63150975539577213875955977149, 4.54338866634520780302783749167, 4.97743752742474768646814718854, 5.70804418276667345505510457016, 5.86080638297475991552836630505, 6.69205257398238111475743537251, 7.08470267100953588111221212276, 7.74352952259069312063347770358, 7.85092908730108193919178449370, 8.413752961397944145466971390681, 8.458968035015643325998359679496, 8.831862100813789181510536703591, 9.314120936575675239669110822531, 9.485047597210104391000956809384, 10.28077010141182019098845748019
