| L(s) = 1 | + 20·2-s + 220·3-s + 1.64e3·4-s + 4.40e3·6-s − 5.79e4·7-s + 9.85e4·8-s − 1.63e5·9-s − 6.18e5·11-s + 3.60e5·12-s − 3.41e6·13-s − 1.15e6·14-s + 6.29e5·16-s − 1.31e6·17-s − 3.26e6·18-s + 5.32e6·19-s − 1.27e7·21-s − 1.23e7·22-s − 5.89e7·23-s + 2.16e7·24-s − 6.82e7·26-s − 4.35e7·27-s − 9.49e7·28-s + 9.41e7·29-s + 2.44e8·31-s + 1.18e8·32-s − 1.35e8·33-s − 2.63e7·34-s + ⋯ |
| L(s) = 1 | + 0.441·2-s + 0.522·3-s + 0.800·4-s + 0.231·6-s − 1.30·7-s + 1.06·8-s − 0.922·9-s − 1.15·11-s + 0.418·12-s − 2.55·13-s − 0.575·14-s + 0.150·16-s − 0.225·17-s − 0.407·18-s + 0.493·19-s − 0.680·21-s − 0.511·22-s − 1.90·23-s + 0.555·24-s − 1.12·26-s − 0.584·27-s − 1.04·28-s + 0.852·29-s + 1.53·31-s + 0.622·32-s − 0.604·33-s − 0.0994·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.9738453833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9738453833\) |
| \(L(\frac{13}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 5 p^{2} T - 155 p^{3} T^{2} - 5 p^{13} T^{3} + p^{22} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 220 T + 23530 p^{2} T^{2} - 220 p^{11} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 57900 T + 660624350 p T^{2} + 57900 p^{11} T^{3} + p^{22} T^{4} \) |
| 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
−15.68664509440942668716342900458, −14.57231140080456349488085587095, −13.97917258859909453982592361837, −13.58358617534176262733253986652, −12.76624056195941359637908900131, −12.09103429721113413941776273738, −11.73784056148531848444201892674, −10.62440052727329407562524466579, −9.829110845704631110431037691593, −9.799233143365132046911277578229, −8.329694288214976881922407059345, −7.80059953915034495774581089177, −6.99335364249493061301882880319, −6.33295537173086322688462956118, −5.27816277417329085781471990121, −4.58857880156771686729865202908, −3.26619008515241644474449539009, −2.55209493283170117722763473654, −2.21358425919128354950138886246, −0.29362508716115768650175800344,
0.29362508716115768650175800344, 2.21358425919128354950138886246, 2.55209493283170117722763473654, 3.26619008515241644474449539009, 4.58857880156771686729865202908, 5.27816277417329085781471990121, 6.33295537173086322688462956118, 6.99335364249493061301882880319, 7.80059953915034495774581089177, 8.329694288214976881922407059345, 9.799233143365132046911277578229, 9.829110845704631110431037691593, 10.62440052727329407562524466579, 11.73784056148531848444201892674, 12.09103429721113413941776273738, 12.76624056195941359637908900131, 13.58358617534176262733253986652, 13.97917258859909453982592361837, 14.57231140080456349488085587095, 15.68664509440942668716342900458
