| L(s) = 1 | − 64·2-s + 3.07e3·4-s − 6.25e3·5-s + 1.40e4·7-s − 1.31e5·8-s + 4.00e5·10-s − 4.21e5·11-s + 1.73e6·13-s − 9.01e5·14-s + 5.24e6·16-s + 6.32e6·17-s − 2.88e7·19-s − 1.92e7·20-s + 2.69e7·22-s + 4.52e7·23-s + 2.92e7·25-s − 1.10e8·26-s + 4.32e7·28-s − 5.82e7·29-s + 4.14e7·31-s − 2.01e8·32-s − 4.04e8·34-s − 8.80e7·35-s + 3.77e8·37-s + 1.84e9·38-s + 8.19e8·40-s + 7.85e8·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.316·7-s − 1.41·8-s + 1.26·10-s − 0.789·11-s + 1.29·13-s − 0.448·14-s + 5/4·16-s + 1.08·17-s − 2.67·19-s − 1.34·20-s + 1.11·22-s + 1.46·23-s + 3/5·25-s − 1.82·26-s + 0.475·28-s − 0.527·29-s + 0.259·31-s − 1.06·32-s − 1.52·34-s − 0.283·35-s + 0.894·37-s + 3.78·38-s + 1.26·40-s + 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 14092 T - 309197214 p T^{2} - 14092 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 421584 T + 305428208086 T^{2} + 421584 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1730524 T + 3917874059118 T^{2} - 1730524 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6323628 T + 73563074016262 T^{2} - 6323628 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 28897400 T + 430951545856038 T^{2} + 28897400 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 45236076 T + 2116464952736398 T^{2} - 45236076 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 58226220 T + 22969270731722158 T^{2} + 58226220 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 41413384 T + 48711797584420926 T^{2} - 41413384 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 377255452 T + 373315553507530302 T^{2} - 377255452 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 785271036 T + 985831391781991606 T^{2} - 785271036 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1452987236 T + 2194695988642931238 T^{2} + 1452987236 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1288127748 T + 4501637759066496382 T^{2} - 1288127748 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 30490836 T + 1934178302201224318 T^{2} - 30490836 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8677102440 T + 78821502141035647318 T^{2} + 8677102440 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1115498764 T - 16676167592870147154 T^{2} - 1115498764 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12673769708 T + \)\(18\!\cdots\!82\)\( T^{2} + 12673769708 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 13799832984 T + \)\(49\!\cdots\!06\)\( T^{2} + 13799832984 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17842079516 T + \)\(53\!\cdots\!18\)\( T^{2} + 17842079516 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12636930320 T + \)\(13\!\cdots\!58\)\( T^{2} + 12636930320 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 41986488924 T + \)\(18\!\cdots\!78\)\( T^{2} + 41986488924 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13208740020 T + \)\(53\!\cdots\!78\)\( T^{2} + 13208740020 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 61787462828 T + \)\(91\!\cdots\!02\)\( T^{2} + 61787462828 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−11.21509550205385062917549360425, −10.94272880801045186560779527504, −10.66992759562284699156927361816, −10.10638493936510959713211406839, −9.199784423880429828185159946677, −8.828807610999855446294977492986, −8.158297943631414959187659175371, −8.083313479680583310129657578487, −7.28611142323228441337610547124, −6.77398609513153560843211265550, −6.04501615165366670623595180136, −5.49390100002729893046788943104, −4.41159192603137487279519731798, −3.93970716904597787896783466277, −2.94169676874312130813528080097, −2.54568305894279432270959899532, −1.40488119672030222588779289555, −1.16847508886889707957782100125, 0, 0,
1.16847508886889707957782100125, 1.40488119672030222588779289555, 2.54568305894279432270959899532, 2.94169676874312130813528080097, 3.93970716904597787896783466277, 4.41159192603137487279519731798, 5.49390100002729893046788943104, 6.04501615165366670623595180136, 6.77398609513153560843211265550, 7.28611142323228441337610547124, 8.083313479680583310129657578487, 8.158297943631414959187659175371, 8.828807610999855446294977492986, 9.199784423880429828185159946677, 10.10638493936510959713211406839, 10.66992759562284699156927361816, 10.94272880801045186560779527504, 11.21509550205385062917549360425
