| L(s) = 1 | + 14·2-s − 1.45e3·3-s − 1.75e3·4-s − 3.12e4·5-s − 2.04e4·6-s − 2.98e4·7-s + 6.27e4·8-s + 1.59e6·9-s − 4.37e5·10-s + 3.81e6·11-s + 2.56e6·12-s − 3.63e7·13-s − 4.17e5·14-s + 4.55e7·15-s − 6.11e7·16-s + 5.62e7·17-s + 2.23e7·18-s + 6.84e7·19-s + 5.48e7·20-s + 4.34e7·21-s + 5.34e7·22-s − 1.81e9·23-s − 9.15e7·24-s + 7.32e8·25-s − 5.08e8·26-s − 1.54e9·27-s + 5.23e7·28-s + ⋯ |
| L(s) = 1 | + 0.154·2-s − 1.15·3-s − 0.214·4-s − 0.894·5-s − 0.178·6-s − 0.0958·7-s + 0.0846·8-s + 9-s − 0.138·10-s + 0.649·11-s + 0.247·12-s − 2.08·13-s − 0.0148·14-s + 1.03·15-s − 0.911·16-s + 0.565·17-s + 0.154·18-s + 0.333·19-s + 0.191·20-s + 0.110·21-s + 0.100·22-s − 2.56·23-s − 0.0977·24-s + 3/5·25-s − 0.322·26-s − 0.769·27-s + 0.0205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{6} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 7 p T + 61 p^{5} T^{2} - 7 p^{14} T^{3} + p^{26} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 29832 T + 182374929374 T^{2} + 29832 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3817328 T + 2259748524514 p T^{2} - 3817328 p^{13} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 36300124 T + 791567276964414 T^{2} + 36300124 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 56248076 T + 18450548115326918 T^{2} - 56248076 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 68436344 T + 65625304252026198 T^{2} - 68436344 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1818683376 T + 1834851693645360046 T^{2} + 1818683376 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5904610708 T + 24529291306539155198 T^{2} - 5904610708 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1370608272 T + 34543494712937920382 T^{2} - 1370608272 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11305158812 T + \)\(48\!\cdots\!74\)\( T^{2} + 11305158812 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 44664863092 T + \)\(19\!\cdots\!82\)\( T^{2} - 44664863092 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11622360680 T + \)\(13\!\cdots\!90\)\( T^{2} - 11622360680 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9081282160 T + \)\(10\!\cdots\!10\)\( T^{2} + 9081282160 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 59746289836 T - \)\(29\!\cdots\!34\)\( T^{2} + 59746289836 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 437570176624 T + \)\(23\!\cdots\!78\)\( T^{2} + 437570176624 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 488101155700 T + \)\(27\!\cdots\!78\)\( T^{2} + 488101155700 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1292041643544 T + \)\(12\!\cdots\!58\)\( T^{2} + 1292041643544 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 865462172224 T + \)\(15\!\cdots\!66\)\( T^{2} - 865462172224 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1447638690556 T + \)\(35\!\cdots\!66\)\( T^{2} + 1447638690556 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 623693717520 T + \)\(58\!\cdots\!78\)\( T^{2} + 623693717520 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3102474245208 T + \)\(15\!\cdots\!58\)\( T^{2} + 3102474245208 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15177596537436 T + \)\(98\!\cdots\!78\)\( T^{2} + 15177596537436 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 23186934522044 T + \)\(24\!\cdots\!38\)\( T^{2} + 23186934522044 p^{13} T^{3} + p^{26} T^{4} \) |
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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
−15.98263364788183431292374442966, −15.28370064811597089338090628396, −14.24259683037774744943048705598, −13.98980650928520953437613514301, −12.55292676045391638575532383720, −12.17127410245836013364647414803, −11.84464857501036667510729381255, −10.96431886209760104438709446756, −10.00257865097342878420092435605, −9.563417648981332521232866351628, −8.187064796832874664891637914962, −7.43174722655183591167946179390, −6.66522155311018710480038008560, −5.72857863382139211826320456425, −4.53370029471099188844313519281, −4.36461984440029014702315036985, −2.83990430929045962321671871266, −1.47323106710242416702872637975, 0, 0,
1.47323106710242416702872637975, 2.83990430929045962321671871266, 4.36461984440029014702315036985, 4.53370029471099188844313519281, 5.72857863382139211826320456425, 6.66522155311018710480038008560, 7.43174722655183591167946179390, 8.187064796832874664891637914962, 9.563417648981332521232866351628, 10.00257865097342878420092435605, 10.96431886209760104438709446756, 11.84464857501036667510729381255, 12.17127410245836013364647414803, 12.55292676045391638575532383720, 13.98980650928520953437613514301, 14.24259683037774744943048705598, 15.28370064811597089338090628396, 15.98263364788183431292374442966
