L(s) = 1 | + 2-s + 3-s + 8·5-s + 6-s − 7-s − 8-s + 8·10-s + 3·11-s − 5·13-s − 14-s + 8·15-s − 16-s + 5·17-s − 3·19-s − 21-s + 3·22-s − 6·23-s − 24-s + 38·25-s − 5·26-s − 27-s + 9·29-s + 8·30-s − 8·31-s + 3·33-s + 5·34-s − 8·35-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 3.57·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 2.52·10-s + 0.904·11-s − 1.38·13-s − 0.267·14-s + 2.06·15-s − 1/4·16-s + 1.21·17-s − 0.688·19-s − 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s + 38/5·25-s − 0.980·26-s − 0.192·27-s + 1.67·29-s + 1.46·30-s − 1.43·31-s + 0.522·33-s + 0.857·34-s − 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.528997866\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.528997866\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.65234761240300713044289414016, −10.48036719555506605811388077778, −9.909368445830808424276765756360, −9.702530217671609123200659693485, −9.387213486985284318908274632670, −9.182987224855806064294159330136, −8.503923422612367660369418439499, −8.013122467384839701530357397457, −7.23088041632629343234814940258, −6.51905856959987526073644511782, −6.32663603249249366271002008719, −6.12047470895864318007447049586, −5.40245095979644550487900414957, −4.98948553865467236865492509446, −4.77580161198113373617957070828, −3.56419567736335344125202481975, −3.15942315338852056502104693645, −2.32812910948998003646911401521, −2.06651423055000875473771703220, −1.41657840200847134643381722163,
1.41657840200847134643381722163, 2.06651423055000875473771703220, 2.32812910948998003646911401521, 3.15942315338852056502104693645, 3.56419567736335344125202481975, 4.77580161198113373617957070828, 4.98948553865467236865492509446, 5.40245095979644550487900414957, 6.12047470895864318007447049586, 6.32663603249249366271002008719, 6.51905856959987526073644511782, 7.23088041632629343234814940258, 8.013122467384839701530357397457, 8.503923422612367660369418439499, 9.182987224855806064294159330136, 9.387213486985284318908274632670, 9.702530217671609123200659693485, 9.909368445830808424276765756360, 10.48036719555506605811388077778, 10.65234761240300713044289414016