| L(s) = 1 | + 3·3-s − 8·4-s + 18·7-s + 90·11-s − 24·12-s − 65·13-s + 117·17-s − 42·19-s + 54·21-s − 18·23-s + 223·25-s − 27·27-s − 144·28-s + 99·29-s + 270·33-s + 195·37-s − 195·39-s − 63·41-s + 82·43-s − 720·44-s − 127·49-s + 351·51-s + 520·52-s − 522·53-s − 126·57-s − 1.36e3·59-s + 719·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.971·7-s + 2.46·11-s − 0.577·12-s − 1.38·13-s + 1.66·17-s − 0.507·19-s + 0.561·21-s − 0.163·23-s + 1.78·25-s − 0.192·27-s − 0.971·28-s + 0.633·29-s + 1.42·33-s + 0.866·37-s − 0.800·39-s − 0.239·41-s + 0.290·43-s − 2.46·44-s − 0.370·49-s + 0.963·51-s + 1.38·52-s − 1.35·53-s − 0.292·57-s − 3.01·59-s + 1.50·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.708736275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708736275\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 223 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 18 T + 451 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 90 T + 4031 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 117 T + 8776 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 42 T + 7447 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T - 11843 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 99 T - 14588 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 21950 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 195 T + 63328 T^{2} - 195 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 63 T + 70244 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 82 T - 72783 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 202354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 261 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1368 T + 829187 T^{2} + 1368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 901 T + p^{3} T^{2} )( 1 + 182 T + p^{3} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 1218 T + 795271 T^{2} + 1218 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 810 T + 576611 T^{2} + 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 309959 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 440 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 284726 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2628 T + 3007097 T^{2} - 2628 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2004 T + 2251345 T^{2} + 2004 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−16.19070696740159999958382328021, −14.95312518381551538990764003931, −14.54893601012994124085389315948, −14.46042472992647119789468202108, −14.04472311041188170855034220425, −13.17085211965414627495141601243, −12.25832657975772405799141516690, −12.09929814719852620392873192070, −11.30467924244811007145418504708, −10.40673653615288329941184404978, −9.525540148482312574959661327584, −9.225365385306053188226397977005, −8.574378121638728846891101503349, −7.86280362991122280341051141812, −7.04493677831332736931288409360, −6.09233237928203958508283199163, −4.77684931404360050921121243664, −4.42164068468694508024683721833, −3.17206571869895288643449316524, −1.36087313542375749807834651046,
1.36087313542375749807834651046, 3.17206571869895288643449316524, 4.42164068468694508024683721833, 4.77684931404360050921121243664, 6.09233237928203958508283199163, 7.04493677831332736931288409360, 7.86280362991122280341051141812, 8.574378121638728846891101503349, 9.225365385306053188226397977005, 9.525540148482312574959661327584, 10.40673653615288329941184404978, 11.30467924244811007145418504708, 12.09929814719852620392873192070, 12.25832657975772405799141516690, 13.17085211965414627495141601243, 14.04472311041188170855034220425, 14.46042472992647119789468202108, 14.54893601012994124085389315948, 14.95312518381551538990764003931, 16.19070696740159999958382328021
