| L(s) = 1 | + 2·3-s − 8·5-s + 14·7-s − 6·9-s − 22·11-s − 18·13-s − 16·15-s − 74·17-s − 160·19-s + 28·21-s + 20·23-s − 182·25-s + 22·27-s + 72·29-s − 238·31-s − 44·33-s − 112·35-s − 56·37-s − 36·39-s + 262·41-s + 376·43-s + 48·45-s + 78·47-s + 147·49-s − 148·51-s + 496·53-s + 176·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.715·5-s + 0.755·7-s − 2/9·9-s − 0.603·11-s − 0.384·13-s − 0.275·15-s − 1.05·17-s − 1.93·19-s + 0.290·21-s + 0.181·23-s − 1.45·25-s + 0.156·27-s + 0.461·29-s − 1.37·31-s − 0.232·33-s − 0.540·35-s − 0.248·37-s − 0.147·39-s + 0.997·41-s + 1.33·43-s + 0.159·45-s + 0.242·47-s + 3/7·49-s − 0.406·51-s + 1.28·53-s + 0.431·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 8 T + 246 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 18 T + 3030 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 74 T + 8070 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 160 T + 20038 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 20 T + 19934 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 72 T + 30854 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 238 T + 73338 T^{2} + 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 56 T - 42410 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 262 T + 128358 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 376 T + 152038 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 78 T + 171322 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 496 T + 358758 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 998 T + 620154 T^{2} + 998 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1562 T + 1063518 T^{2} - 1562 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 500 T + 658246 T^{2} - 500 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2100 T + 1817342 T^{2} + 2100 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 586 T + 43758 T^{2} - 586 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 985422 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1648 T + 1419270 T^{2} - 1648 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 940 T + 1238838 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1152 T + 2072622 T^{2} - 1152 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.959972410253618877614663164912, −8.766682748563349426969670543923, −8.197669088511216850279697691680, −8.097049928413454484847070384127, −7.46976553004778758148066027127, −7.28692272837988702616436077594, −6.73993681424293674805772528041, −6.22311741924035161602309228244, −5.68962961358465638430633337136, −5.40345100893646698679185188488, −4.61738634343120990046799781716, −4.41699164947124921496560406802, −3.92947039733666461026610878244, −3.57671291501064408563487291310, −2.56342755547118177495019595074, −2.39328452904396865660747681450, −1.93618153059689036238298678767, −1.04459560422884113672584049805, 0, 0,
1.04459560422884113672584049805, 1.93618153059689036238298678767, 2.39328452904396865660747681450, 2.56342755547118177495019595074, 3.57671291501064408563487291310, 3.92947039733666461026610878244, 4.41699164947124921496560406802, 4.61738634343120990046799781716, 5.40345100893646698679185188488, 5.68962961358465638430633337136, 6.22311741924035161602309228244, 6.73993681424293674805772528041, 7.28692272837988702616436077594, 7.46976553004778758148066027127, 8.097049928413454484847070384127, 8.197669088511216850279697691680, 8.766682748563349426969670543923, 8.959972410253618877614663164912
