| L(s) = 1 | − 3·3-s + 2·5-s + 2·7-s + 6·9-s − 32·11-s + 27·13-s − 6·15-s − 22·17-s − 38·19-s − 6·21-s + 40·23-s + 25·25-s − 9·27-s + 30·29-s + 96·33-s + 4·35-s − 81·39-s − 24·41-s − 49·43-s + 12·45-s + 46·47-s − 95·49-s + 66·51-s − 84·53-s − 64·55-s + 114·57-s + 114·59-s + ⋯ |
| L(s) = 1 | − 3-s + 2/5·5-s + 2/7·7-s + 2/3·9-s − 2.90·11-s + 2.07·13-s − 2/5·15-s − 1.29·17-s − 2·19-s − 2/7·21-s + 1.73·23-s + 25-s − 1/3·27-s + 1.03·29-s + 2.90·33-s + 4/35·35-s − 2.07·39-s − 0.585·41-s − 1.13·43-s + 4/15·45-s + 0.978·47-s − 1.93·49-s + 1.29·51-s − 1.58·53-s − 1.16·55-s + 2·57-s + 1.93·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.110061078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110061078\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 27 T + 412 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T + 195 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T + 1071 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 30 T + 1141 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 601 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2495 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 24 T + 1873 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 49 T + 552 T^{2} + 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 46 T - 93 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 84 T + 5161 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T + 7813 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 97 T + 5688 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 45 T + 5164 T^{2} - 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 84 T + 7393 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 35 T - 4104 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 66 T + 9373 T^{2} + 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 108 T + 13297 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.24663271245405403042240815131, −9.974818738230425908094484805558, −9.162395086499320923245610504022, −8.767915940042164497238343812642, −8.297527240441737650583372561174, −8.224230624011749101885159893770, −7.67798130070173116229662946795, −6.76004682635692444977232427449, −6.58195562829461375867208906795, −6.44528218927577536328470491133, −5.70980782495832438522797120028, −5.13329655323861776094303914536, −4.95156242445396215273711041273, −4.67535189614147769874615993908, −3.76998426192481984713727362576, −3.24806116852241526100197985956, −2.40105091743757672843814327720, −2.17894078228272399717872110227, −1.13856745598346454833444355645, −0.40289497913648998223260979410,
0.40289497913648998223260979410, 1.13856745598346454833444355645, 2.17894078228272399717872110227, 2.40105091743757672843814327720, 3.24806116852241526100197985956, 3.76998426192481984713727362576, 4.67535189614147769874615993908, 4.95156242445396215273711041273, 5.13329655323861776094303914536, 5.70980782495832438522797120028, 6.44528218927577536328470491133, 6.58195562829461375867208906795, 6.76004682635692444977232427449, 7.67798130070173116229662946795, 8.224230624011749101885159893770, 8.297527240441737650583372561174, 8.767915940042164497238343812642, 9.162395086499320923245610504022, 9.974818738230425908094484805558, 10.24663271245405403042240815131
