| L(s) = 1 | + 16·2-s + 30·3-s − 20·4-s − 250·5-s + 480·6-s − 2.68e3·8-s − 2.11e3·9-s − 4.00e3·10-s − 7.90e3·11-s − 600·12-s + 1.78e4·13-s − 7.50e3·15-s − 2.11e4·16-s + 2.39e3·17-s − 3.38e4·18-s + 3.61e3·19-s + 5.00e3·20-s − 1.26e5·22-s + 1.38e4·23-s − 8.06e4·24-s + 4.68e4·25-s + 2.85e5·26-s − 8.82e4·27-s − 1.26e5·29-s − 1.20e5·30-s − 2.52e5·31-s + 1.01e5·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.641·3-s − 0.156·4-s − 0.894·5-s + 0.907·6-s − 1.85·8-s − 0.967·9-s − 1.26·10-s − 1.79·11-s − 0.100·12-s + 2.24·13-s − 0.573·15-s − 1.28·16-s + 0.118·17-s − 1.36·18-s + 0.120·19-s + 0.139·20-s − 2.53·22-s + 0.237·23-s − 1.19·24-s + 3/5·25-s + 3.18·26-s − 0.863·27-s − 0.966·29-s − 0.811·30-s − 1.52·31-s + 0.545·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p^{4} T + 69 p^{2} T^{2} - p^{11} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 10 p T + 335 p^{2} T^{2} - 10 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7906 T + 48246951 T^{2} + 7906 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 17818 T + 197179459 T^{2} - 17818 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2398 T + 607214547 T^{2} - 2398 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3612 T + 439599498 T^{2} - 3612 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 13844 T + 1829416578 T^{2} - 13844 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 126898 T + 13660776923 T^{2} + 126898 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 252768 T + 48616095374 T^{2} + 252768 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 265860 T + 123746037230 T^{2} + 265860 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 111920 T + 331245931458 T^{2} - 111920 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 947572 T + 741530960314 T^{2} - 947572 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 271274 T + 889978835879 T^{2} + 271274 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1267792 T + 2564062930746 T^{2} + 1267792 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1360120 T + 4158592150838 T^{2} - 1360120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1813680 T + 7102192237178 T^{2} - 1813680 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2189312 T + 2038788130198 T^{2} + 2189312 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1494928 T + 16579508918702 T^{2} + 1494928 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7169788 T + 32961331353526 T^{2} + 7169788 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7942974 T + 52633585962783 T^{2} + 7942974 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 304712 T + 15462893048006 T^{2} - 304712 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 17943528 T + 166259116359218 T^{2} - 17943528 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4258074 T + 145736945716539 T^{2} + 4258074 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.67205205518446551687050903616, −10.40245779792808271255064122772, −9.365520923740203492880826972928, −8.999096171941555696403927971246, −8.518507976071013967309689339476, −8.401597647680997863965432298130, −7.55457293033431798851249142690, −7.36944671174479846323616262474, −6.08559483233920949823599905826, −5.93972164605400222172369886970, −5.33197664872599479060165726047, −4.97936066646366575012971873984, −4.08758724439619261561065962108, −3.84308595001482048764038650155, −3.13226269777334553990004109191, −3.07726771638768820197366064842, −2.10538652732723668076704295305, −1.04479851592560756619512963811, 0, 0,
1.04479851592560756619512963811, 2.10538652732723668076704295305, 3.07726771638768820197366064842, 3.13226269777334553990004109191, 3.84308595001482048764038650155, 4.08758724439619261561065962108, 4.97936066646366575012971873984, 5.33197664872599479060165726047, 5.93972164605400222172369886970, 6.08559483233920949823599905826, 7.36944671174479846323616262474, 7.55457293033431798851249142690, 8.401597647680997863965432298130, 8.518507976071013967309689339476, 8.999096171941555696403927971246, 9.365520923740203492880826972928, 10.40245779792808271255064122772, 10.67205205518446551687050903616
