| L(s) = 1 | + 16·3-s − 16·5-s + 243·9-s + 76·11-s + 1.76e3·13-s − 256·15-s + 1.05e3·17-s − 1.93e3·19-s − 936·23-s + 3.12e3·25-s + 7.56e3·27-s − 7.96e3·29-s − 1.56e3·31-s + 1.21e3·33-s − 4.93e3·37-s + 2.81e4·39-s − 3.16e4·41-s − 3.28e4·43-s − 3.88e3·45-s + 2.07e4·47-s + 1.68e4·51-s + 3.74e4·53-s − 1.21e3·55-s − 3.09e4·57-s − 2.11e4·59-s + 2.99e3·61-s − 2.81e4·65-s + ⋯ |
| L(s) = 1 | + 1.02·3-s − 0.286·5-s + 9-s + 0.189·11-s + 2.88·13-s − 0.293·15-s + 0.886·17-s − 1.23·19-s − 0.368·23-s + 25-s + 1.99·27-s − 1.75·29-s − 0.293·31-s + 0.194·33-s − 0.592·37-s + 2.96·39-s − 2.94·41-s − 2.70·43-s − 0.286·45-s + 1.37·47-s + 0.909·51-s + 1.82·53-s − 0.0542·55-s − 1.26·57-s − 0.790·59-s + 0.102·61-s − 0.826·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.943175905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.943175905\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 16 T + 13 T^{2} - 16 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 16 T - 2869 T^{2} + 16 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 76 T - 155275 T^{2} - 76 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 880 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1056 T - 304721 T^{2} - 1056 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 1936 T + 1271997 T^{2} + 1936 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 936 T - 5560247 T^{2} + 936 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3982 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 1568 T - 26170527 T^{2} + 1568 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4938 T - 44960113 T^{2} + 4938 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 15840 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 16412 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 20768 T + 201964817 T^{2} - 20768 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 37402 T + 980714111 T^{2} - 37402 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 21136 T - 268193803 T^{2} + 21136 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2992 T - 835644237 T^{2} - 2992 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 45836 T + 750813789 T^{2} - 45836 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 49840 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 56320 T + 1098870807 T^{2} - 56320 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 40744 T - 1416982863 T^{2} + 40744 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 112464 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 64256 T - 1455225913 T^{2} + 64256 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2272 T + p^{5} T^{2} )^{2} \) |
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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
−11.81561721452964933516469377766, −11.40397241812851182311984451600, −10.69561159184162142957522118339, −10.41391690129658148311588028599, −10.03886602568073940224929296641, −8.942898202073100229609197807777, −8.913900183961893331339833135641, −8.337777881192578572905154500694, −8.169400650125239181379762570001, −7.23631401458945937679305316037, −6.73325904190581035123164386360, −6.30201142062489954899870952425, −5.55214891417132274776228048930, −4.83078896996562077526495690341, −3.97068208298486547358754977853, −3.43351682167006167427289163902, −3.34176579791895165297360408445, −1.95317081922379601423345697329, −1.53492376867214420360106214468, −0.66966273850264665563748043896,
0.66966273850264665563748043896, 1.53492376867214420360106214468, 1.95317081922379601423345697329, 3.34176579791895165297360408445, 3.43351682167006167427289163902, 3.97068208298486547358754977853, 4.83078896996562077526495690341, 5.55214891417132274776228048930, 6.30201142062489954899870952425, 6.73325904190581035123164386360, 7.23631401458945937679305316037, 8.169400650125239181379762570001, 8.337777881192578572905154500694, 8.913900183961893331339833135641, 8.942898202073100229609197807777, 10.03886602568073940224929296641, 10.41391690129658148311588028599, 10.69561159184162142957522118339, 11.40397241812851182311984451600, 11.81561721452964933516469377766
