| L(s) = 1 | − 20·2-s + 220·3-s + 1.64e3·4-s + 6.25e3·5-s − 4.40e3·6-s − 9.85e4·8-s − 1.63e5·9-s − 1.25e5·10-s − 6.18e5·11-s + 3.60e5·12-s − 3.41e6·13-s + 1.37e6·15-s + 6.29e5·16-s − 1.31e6·17-s + 3.26e6·18-s − 5.32e6·19-s + 1.02e7·20-s + 1.23e7·22-s + 5.89e7·23-s − 2.16e7·24-s + 2.92e7·25-s + 6.82e7·26-s − 4.35e7·27-s + 9.41e7·29-s − 2.75e7·30-s − 2.44e8·31-s − 1.18e8·32-s + ⋯ |
| L(s) = 1 | − 0.441·2-s + 0.522·3-s + 0.800·4-s + 0.894·5-s − 0.231·6-s − 1.06·8-s − 0.922·9-s − 0.395·10-s − 1.15·11-s + 0.418·12-s − 2.55·13-s + 0.467·15-s + 0.150·16-s − 0.225·17-s + 0.407·18-s − 0.493·19-s + 0.716·20-s + 0.511·22-s + 1.90·23-s − 0.555·24-s + 3/5·25-s + 1.12·26-s − 0.584·27-s + 0.852·29-s − 0.206·30-s − 1.53·31-s − 0.622·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.683122549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.683122549\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 5 p^{2} T - 155 p^{3} T^{2} + 5 p^{13} T^{3} + p^{22} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 220 T + 23530 p^{2} T^{2} - 220 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 618176 T + 245194892966 T^{2} + 618176 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3414260 T + 6398662197390 T^{2} + 3414260 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1317940 T + 59308395866630 T^{2} + 1317940 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 280280 p T + 194538827137638 T^{2} + 280280 p^{12} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2562780 p T + 2773540471931410 T^{2} - 2562780 p^{12} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3246220 p T + 23426350431097358 T^{2} - 3246220 p^{12} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 244543464 T + 34393316207729486 T^{2} + 244543464 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 21003220 T + 137126715218410590 T^{2} - 21003220 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 745743316 T + 929792912462405846 T^{2} - 745743316 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 629950100 T + 1840945003918927050 T^{2} - 629950100 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1402061540 T + 5181805952108806370 T^{2} - 1402061540 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1138320580 T - 2203723231625575330 T^{2} - 1138320580 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7317515560 T + 55027608950440780118 T^{2} + 7317515560 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1516425676 T + 85869525433683691566 T^{2} - 1516425676 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 15734290140 T + \)\(30\!\cdots\!30\)\( T^{2} - 15734290140 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} - 32938471544 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 29982848860 T + \)\(78\!\cdots\!10\)\( T^{2} - 29982848860 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} + 3302823120 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13299102420 T + \)\(18\!\cdots\!30\)\( T^{2} + 13299102420 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} - 12674770860 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3080703740 T + \)\(18\!\cdots\!70\)\( T^{2} - 3080703740 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−10.59518901828211937165148394180, −9.649524967431193453744246113398, −9.303255475153339094267202884893, −9.256130100963732513939868466190, −8.487631451250251431369362547059, −8.026445738894752591603836490865, −7.46355850028373552754832820108, −6.95693002571773374783413553515, −6.72426997552573734880728477204, −5.88478673466463548909009366982, −5.43734713114021171142596544216, −5.06998644824891361109353978233, −4.53970908346155193040331382748, −3.46506680092614851900983512190, −2.81869101621283856970275112921, −2.44620412736602135786798126530, −2.44487240150485798677161163977, −1.79021339261273909024440086227, −0.62196683767690785674218394785, −0.44516493932278026984765769594,
0.44516493932278026984765769594, 0.62196683767690785674218394785, 1.79021339261273909024440086227, 2.44487240150485798677161163977, 2.44620412736602135786798126530, 2.81869101621283856970275112921, 3.46506680092614851900983512190, 4.53970908346155193040331382748, 5.06998644824891361109353978233, 5.43734713114021171142596544216, 5.88478673466463548909009366982, 6.72426997552573734880728477204, 6.95693002571773374783413553515, 7.46355850028373552754832820108, 8.026445738894752591603836490865, 8.487631451250251431369362547059, 9.256130100963732513939868466190, 9.303255475153339094267202884893, 9.649524967431193453744246113398, 10.59518901828211937165148394180
