| L(s) = 1 | − 2-s − 54·3-s − 37·4-s − 194·5-s + 54·6-s + 418·7-s − 53·8-s + 2.18e3·9-s + 194·10-s + 1.99e3·12-s + 1.32e4·13-s − 418·14-s + 1.04e4·15-s − 1.47e4·16-s + 1.02e4·17-s − 2.18e3·18-s − 1.41e4·19-s + 7.17e3·20-s − 2.25e4·21-s − 1.36e4·23-s + 2.86e3·24-s − 8.52e4·25-s − 1.32e4·26-s − 7.87e4·27-s − 1.54e4·28-s − 1.53e4·29-s − 1.04e4·30-s + ⋯ |
| L(s) = 1 | − 0.0883·2-s − 1.15·3-s − 0.289·4-s − 0.694·5-s + 0.102·6-s + 0.460·7-s − 0.0365·8-s + 9-s + 0.0613·10-s + 0.333·12-s + 1.67·13-s − 0.0407·14-s + 0.801·15-s − 0.903·16-s + 0.506·17-s − 0.0883·18-s − 0.474·19-s + 0.200·20-s − 0.531·21-s − 0.234·23-s + 0.0422·24-s − 1.09·25-s − 0.147·26-s − 0.769·27-s − 0.133·28-s − 0.116·29-s − 0.0708·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 19 p T^{2} + p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 194 T + 122882 T^{2} + 194 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 418 T + 1305774 T^{2} - 418 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 13246 T + 169046010 T^{2} - 13246 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10256 T + 839902430 T^{2} - 10256 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 14196 T + 610166774 T^{2} + 14196 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 13666 T + 5939145158 T^{2} + 13666 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 528 p T + 30012198502 T^{2} + 528 p^{8} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 48040 T + 55594799550 T^{2} + 48040 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 274092 T + 171168601790 T^{2} - 274092 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 755836 T + 339658819958 T^{2} + 755836 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1704096 T + 1258258595846 T^{2} - 1704096 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1182094 T + 965730056342 T^{2} + 1182094 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2156394 T + 3355666109890 T^{2} + 2156394 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 927332 T + 3919039493894 T^{2} - 927332 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1061994 T + 60938203298 T^{2} - 1061994 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 48656 p T + 14776996218054 T^{2} + 48656 p^{8} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5495514 T + 25737621036694 T^{2} + 5495514 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5450812 T + 26436565990902 T^{2} - 5450812 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1536590 T + 38585225535486 T^{2} - 1536590 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8850888 T + 59626634719558 T^{2} + 8850888 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6810132 T + 98894344907446 T^{2} - 6810132 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9897376 T + 135461286702270 T^{2} - 9897376 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−9.831575662732200264925014262407, −9.813295862725811942173135933071, −9.048981863583937069677555026370, −8.543803637088079091013253804676, −8.168098702536855072249095737833, −7.60073725120106580651801768083, −7.29373831880768657935531334898, −6.38363281126626203757686176815, −6.21062193193040110295799164670, −5.82644494632973049376080832522, −4.96613849025323133041820316219, −4.73573538558698337937196506250, −3.92537741291075408713669047361, −3.84726151605456720412161014896, −2.98444834390194816717550296085, −2.01737878433001876996289705646, −1.45374081928874399684159938440, −0.933041284049226980599718158644, 0, 0,
0.933041284049226980599718158644, 1.45374081928874399684159938440, 2.01737878433001876996289705646, 2.98444834390194816717550296085, 3.84726151605456720412161014896, 3.92537741291075408713669047361, 4.73573538558698337937196506250, 4.96613849025323133041820316219, 5.82644494632973049376080832522, 6.21062193193040110295799164670, 6.38363281126626203757686176815, 7.29373831880768657935531334898, 7.60073725120106580651801768083, 8.168098702536855072249095737833, 8.543803637088079091013253804676, 9.048981863583937069677555026370, 9.813295862725811942173135933071, 9.831575662732200264925014262407
