| L(s) = 1 | + 2-s + 2·3-s + 8·4-s − 16·5-s + 2·6-s + 23·8-s + 27·9-s − 16·10-s + 8·11-s + 16·12-s + 56·13-s − 32·15-s + 23·16-s − 54·17-s + 27·18-s + 110·19-s − 128·20-s + 8·22-s − 48·23-s + 46·24-s + 125·25-s + 56·26-s + 154·27-s − 220·29-s − 32·30-s − 12·31-s + 184·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.384·3-s + 4-s − 1.43·5-s + 0.136·6-s + 1.01·8-s + 9-s − 0.505·10-s + 0.219·11-s + 0.384·12-s + 1.19·13-s − 0.550·15-s + 0.359·16-s − 0.770·17-s + 0.353·18-s + 1.32·19-s − 1.43·20-s + 0.0775·22-s − 0.435·23-s + 0.391·24-s + 25-s + 0.422·26-s + 1.09·27-s − 1.40·29-s − 0.194·30-s − 0.0695·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.516357044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.516357044\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - 7 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T - 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 16 T + 131 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T - 1267 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 54 T - 1997 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 110 T + 5241 T^{2} - 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 48 T - 9863 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 12 T - 29647 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 246 T + 9863 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 324 T + 1153 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 162 T - 122633 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 810 T + 450721 T^{2} + 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 p T + 3 p^{2} T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 244 T - 241227 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 702 T + 103787 T^{2} - 702 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 440 T - 299439 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1302 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 730 T - 172069 T^{2} + 730 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
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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
−15.46889783696523621842747331906, −15.06345607608593443741083207213, −14.22546057348319997644493014164, −13.77265539156942069736713636512, −12.85498676468083956275921505827, −12.78562465886738234565250622269, −11.55427095132005709842121278934, −11.53097860870844103969127630530, −10.94955729157520126930059748839, −10.22456504513501829237877733088, −9.345898579344221918551748717528, −8.581594469756967746335107110100, −7.61513515356107126130760720471, −7.51040439803834874125895702703, −6.72818993492368257719414017129, −5.82706492510803844467686109859, −4.40179371253898927412225609898, −4.04952151809951250425555791784, −2.94662922883036392193526423579, −1.41194297365436200775023785024,
1.41194297365436200775023785024, 2.94662922883036392193526423579, 4.04952151809951250425555791784, 4.40179371253898927412225609898, 5.82706492510803844467686109859, 6.72818993492368257719414017129, 7.51040439803834874125895702703, 7.61513515356107126130760720471, 8.581594469756967746335107110100, 9.345898579344221918551748717528, 10.22456504513501829237877733088, 10.94955729157520126930059748839, 11.53097860870844103969127630530, 11.55427095132005709842121278934, 12.78562465886738234565250622269, 12.85498676468083956275921505827, 13.77265539156942069736713636512, 14.22546057348319997644493014164, 15.06345607608593443741083207213, 15.46889783696523621842747331906
