| L(s) = 1 | + 2-s − 3-s + 6·5-s − 6-s − 2·7-s − 8-s + 6·10-s − 6·11-s − 7·13-s − 2·14-s − 6·15-s − 16-s + 3·17-s − 2·19-s + 2·21-s − 6·22-s + 6·23-s + 24-s + 17·25-s − 7·26-s + 27-s − 3·29-s − 6·30-s − 8·31-s + 6·33-s + 3·34-s − 12·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 2.68·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1.89·10-s − 1.80·11-s − 1.94·13-s − 0.534·14-s − 1.54·15-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.436·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 17/5·25-s − 1.37·26-s + 0.192·27-s − 0.557·29-s − 1.09·30-s − 1.43·31-s + 1.04·33-s + 0.514·34-s − 2.02·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.185151523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.185151523\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−14.35286686074572368409181867512, −14.27867396311595024572434941436, −13.51817964603147958594568172442, −12.98706950235938947359996027931, −12.76007350342529018271265712239, −12.56867437975354614470896300496, −11.46457703036581464797198554723, −10.69811513121485893252081626758, −10.13411025138224755705732726398, −9.948149548125888805029333159835, −9.336707253707236921592227844229, −8.819409285103628851972636338272, −7.36944870964190706456148621822, −7.16326943265698509724527658031, −5.89970760623753785312771422036, −5.63234435803273217669811201665, −5.45151851794710074846736962050, −4.50903381537671760984673464703, −2.71535848121464506175339869070, −2.41599769197848907918756184175,
2.41599769197848907918756184175, 2.71535848121464506175339869070, 4.50903381537671760984673464703, 5.45151851794710074846736962050, 5.63234435803273217669811201665, 5.89970760623753785312771422036, 7.16326943265698509724527658031, 7.36944870964190706456148621822, 8.819409285103628851972636338272, 9.336707253707236921592227844229, 9.948149548125888805029333159835, 10.13411025138224755705732726398, 10.69811513121485893252081626758, 11.46457703036581464797198554723, 12.56867437975354614470896300496, 12.76007350342529018271265712239, 12.98706950235938947359996027931, 13.51817964603147958594568172442, 14.27867396311595024572434941436, 14.35286686074572368409181867512
