| L(s) = 1 | + 9·3-s − 34·5-s + 332·11-s + 2.05e3·13-s − 306·15-s + 922·17-s + 452·19-s + 3.77e3·23-s + 3.12e3·25-s − 729·27-s + 2.33e3·29-s − 9.79e3·31-s + 2.98e3·33-s − 2.39e3·37-s + 1.84e4·39-s + 1.44e4·41-s + 9.30e3·43-s + 2.46e4·47-s + 8.29e3·51-s − 1.11e3·53-s − 1.12e4·55-s + 4.06e3·57-s + 4.68e4·59-s − 9.76e3·61-s − 6.97e4·65-s + 2.62e4·67-s + 3.39e4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.608·5-s + 0.827·11-s + 3.36·13-s − 0.351·15-s + 0.773·17-s + 0.287·19-s + 1.48·23-s + 25-s − 0.192·27-s + 0.514·29-s − 1.83·31-s + 0.477·33-s − 0.287·37-s + 1.94·39-s + 1.34·41-s + 0.767·43-s + 1.62·47-s + 0.446·51-s − 0.0542·53-s − 0.503·55-s + 0.165·57-s + 1.75·59-s − 0.335·61-s − 2.04·65-s + 0.714·67-s + 0.859·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.516932953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.516932953\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 34 T - 1969 T^{2} + 34 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 332 T - 50827 T^{2} - 332 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 1026 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 922 T - 569773 T^{2} - 922 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 452 T - 2271795 T^{2} - 452 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3776 T + 7821833 T^{2} - 3776 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1166 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 9792 T + 67254113 T^{2} + 9792 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2390 T - 63631857 T^{2} + 2390 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7230 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4652 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 24672 T + 379362577 T^{2} - 24672 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1110 T - 416963393 T^{2} + 1110 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 46892 T + 1483935365 T^{2} - 46892 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9762 T - 749299657 T^{2} + 9762 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26252 T - 660957603 T^{2} - 26252 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 65440 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 5606 T - 2041644357 T^{2} + 5606 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 9840 T - 2980230799 T^{2} - 9840 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 61108 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 62958 T - 1620349685 T^{2} + 62958 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 37838 T + p^{5} T^{2} )^{2} \) |
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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
−10.20416110408918513718029922972, −9.434824891745122419984065294869, −9.126141365531566457855009907065, −8.778074570163057877507852833064, −8.498396727634016014894141175242, −8.096891387053096270548866263471, −7.43608506073343121737789303452, −7.10274141324486456222338863923, −6.53475247485694311099989551014, −6.12644637395723930757046707358, −5.54117874671356423701372230269, −5.22168825446719444274542194233, −4.20894398792025486511339071699, −3.84632088306085466294044263234, −3.60211630900326925667652027265, −3.09821921420994812021126176586, −2.38595868765742204590771505893, −1.44321178541878259114200827147, −0.995619255154840588672015310130, −0.75944980829321898402086288270,
0.75944980829321898402086288270, 0.995619255154840588672015310130, 1.44321178541878259114200827147, 2.38595868765742204590771505893, 3.09821921420994812021126176586, 3.60211630900326925667652027265, 3.84632088306085466294044263234, 4.20894398792025486511339071699, 5.22168825446719444274542194233, 5.54117874671356423701372230269, 6.12644637395723930757046707358, 6.53475247485694311099989551014, 7.10274141324486456222338863923, 7.43608506073343121737789303452, 8.096891387053096270548866263471, 8.498396727634016014894141175242, 8.778074570163057877507852833064, 9.126141365531566457855009907065, 9.434824891745122419984065294869, 10.20416110408918513718029922972
