| L(s) = 1 | − 14·2-s − 1.75e3·4-s + 3.12e4·5-s − 2.98e4·7-s − 6.27e4·8-s − 4.37e5·10-s − 3.81e6·11-s − 3.63e7·13-s + 4.17e5·14-s − 6.11e7·16-s − 5.62e7·17-s + 6.84e7·19-s − 5.48e7·20-s + 5.34e7·22-s + 1.81e9·23-s + 7.32e8·25-s + 5.08e8·26-s + 5.23e7·28-s − 5.90e9·29-s + 1.37e9·31-s + 2.12e9·32-s + 7.87e8·34-s − 9.32e8·35-s − 1.13e10·37-s − 9.58e8·38-s − 1.96e9·40-s − 4.46e10·41-s + ⋯ |
| L(s) = 1 | − 0.154·2-s − 0.214·4-s + 0.894·5-s − 0.0958·7-s − 0.0846·8-s − 0.138·10-s − 0.649·11-s − 2.08·13-s + 0.0148·14-s − 0.911·16-s − 0.565·17-s + 0.333·19-s − 0.191·20-s + 0.100·22-s + 2.56·23-s + 3/5·25-s + 0.322·26-s + 0.0205·28-s − 1.84·29-s + 0.277·31-s + 0.349·32-s + 0.0874·34-s − 0.0857·35-s − 0.724·37-s − 0.0516·38-s − 0.0757·40-s − 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 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
−12.89903044016023179811852591780, −12.16300172478606658040825561051, −11.50412081997549806836258446822, −10.85547172816525826052050514124, −10.24221590799233470825757659682, −9.623615975562034921164157261652, −9.160146396494359572424974346712, −8.734799578814202917533237547248, −7.60259953958222395990640146872, −7.14242074623649010823841145429, −6.61167652651255242369066443948, −5.58841410605051434435153610072, −4.90115413950190234090256240282, −4.76002078793005937588611846203, −3.33481373372877103895181306410, −2.66357031320968528850502849640, −2.11055799176691520527033953671, −1.28584268750878585134808407145, 0, 0,
1.28584268750878585134808407145, 2.11055799176691520527033953671, 2.66357031320968528850502849640, 3.33481373372877103895181306410, 4.76002078793005937588611846203, 4.90115413950190234090256240282, 5.58841410605051434435153610072, 6.61167652651255242369066443948, 7.14242074623649010823841145429, 7.60259953958222395990640146872, 8.734799578814202917533237547248, 9.160146396494359572424974346712, 9.623615975562034921164157261652, 10.24221590799233470825757659682, 10.85547172816525826052050514124, 11.50412081997549806836258446822, 12.16300172478606658040825561051, 12.89903044016023179811852591780
