| L(s) = 1 | − 208·2-s − 1.28e4·4-s − 1.56e5·5-s + 1.95e5·7-s + 7.53e6·8-s + 3.25e7·10-s + 1.26e7·11-s − 2.72e8·13-s − 4.07e7·14-s − 5.01e8·16-s + 2.70e9·17-s − 2.24e9·19-s + 2.01e9·20-s − 2.63e9·22-s + 1.62e10·23-s + 1.83e10·25-s + 5.66e10·26-s − 2.51e9·28-s + 1.67e11·29-s − 2.44e11·31-s − 7.62e9·32-s − 5.63e11·34-s − 3.05e10·35-s + 1.05e12·37-s + 4.67e11·38-s − 1.17e12·40-s + 8.11e10·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.392·4-s − 0.894·5-s + 0.0898·7-s + 1.27·8-s + 1.02·10-s + 0.196·11-s − 1.20·13-s − 0.103·14-s − 0.466·16-s + 1.60·17-s − 0.576·19-s + 0.351·20-s − 0.225·22-s + 0.992·23-s + 3/5·25-s + 1.38·26-s − 0.0352·28-s + 1.80·29-s − 1.59·31-s − 0.0392·32-s − 1.83·34-s − 0.0803·35-s + 1.82·37-s + 0.662·38-s − 1.13·40-s + 0.0650·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{7} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 13 p^{4} T + 877 p^{6} T^{2} + 13 p^{19} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 27968 p T + 10537896002 p^{3} T^{2} - 27968 p^{16} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 12677912 T + 8233055677242454 T^{2} - 12677912 p^{15} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 272571732 T + 9120511586549542 p T^{2} + 272571732 p^{15} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2709086588 T + 7398134051960456422 T^{2} - 2709086588 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2248025064 T + 25566844129032196678 T^{2} + 2248025064 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 16214470848 T + \)\(28\!\cdots\!14\)\( T^{2} - 16214470848 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 167604104788 T + \)\(20\!\cdots\!58\)\( T^{2} - 167604104788 p^{15} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 244622120128 T + \)\(39\!\cdots\!02\)\( T^{2} + 244622120128 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 1055700361276 T + \)\(76\!\cdots\!06\)\( T^{2} - 1055700361276 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 81131618188 T + \)\(30\!\cdots\!22\)\( T^{2} - 81131618188 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1312122690840 T + \)\(37\!\cdots\!50\)\( T^{2} - 1312122690840 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7338061052080 T + \)\(37\!\cdots\!90\)\( T^{2} + 7338061052080 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3755998579268 T - \)\(48\!\cdots\!26\)\( T^{2} - 3755998579268 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11942083041176 T + \)\(31\!\cdots\!98\)\( T^{2} - 11942083041176 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18080697194580 T + \)\(47\!\cdots\!98\)\( T^{2} + 18080697194580 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 50227778651128 T + \)\(31\!\cdots\!82\)\( T^{2} + 50227778651128 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 66695913482384 T + \)\(11\!\cdots\!66\)\( T^{2} + 66695913482384 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 232554112724748 T + \)\(29\!\cdots\!54\)\( T^{2} + 232554112724748 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 161119142079040 T + \)\(24\!\cdots\!98\)\( T^{2} + 161119142079040 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 154296276009096 T + \)\(12\!\cdots\!42\)\( T^{2} + 154296276009096 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 305587106859156 T + \)\(30\!\cdots\!78\)\( T^{2} + 305587106859156 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1018905479409788 T + \)\(10\!\cdots\!22\)\( T^{2} + 1018905479409788 p^{15} T^{3} + p^{30} T^{4} \) |
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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
−12.11566838186852333959412885853, −11.81361849920849091590505077386, −10.96423312325528465691185728276, −10.40612661444654519008621827844, −9.655155830102422139161584667977, −9.530697295508529443935504555761, −8.515056349323847306041735008740, −8.380207922927890825888143010726, −7.44789870430171950536792528911, −7.35735676284427617804606511288, −6.25721118984210530928275309570, −5.33791632357187426136756325585, −4.60593991234589321818442495302, −4.26205344495387487392988542031, −3.20859586813912989139021952385, −2.68899234681309744666094846499, −1.36948615926875473074753111394, −1.03141300074246998001052645487, 0, 0,
1.03141300074246998001052645487, 1.36948615926875473074753111394, 2.68899234681309744666094846499, 3.20859586813912989139021952385, 4.26205344495387487392988542031, 4.60593991234589321818442495302, 5.33791632357187426136756325585, 6.25721118984210530928275309570, 7.35735676284427617804606511288, 7.44789870430171950536792528911, 8.380207922927890825888143010726, 8.515056349323847306041735008740, 9.530697295508529443935504555761, 9.655155830102422139161584667977, 10.40612661444654519008621827844, 10.96423312325528465691185728276, 11.81361849920849091590505077386, 12.11566838186852333959412885853
