| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 2·7-s − 4·8-s − 4·10-s + 2·11-s + 4·13-s + 4·14-s + 5·16-s + 2·17-s − 6·19-s + 6·20-s − 4·22-s − 6·23-s + 3·25-s − 8·26-s − 6·28-s − 6·31-s − 6·32-s − 4·34-s − 4·35-s + 6·37-s + 12·38-s − 8·40-s − 10·41-s − 2·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s − 1.41·8-s − 1.26·10-s + 0.603·11-s + 1.10·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s − 1.37·19-s + 1.34·20-s − 0.852·22-s − 1.25·23-s + 3/5·25-s − 1.56·26-s − 1.13·28-s − 1.07·31-s − 1.06·32-s − 0.685·34-s − 0.676·35-s + 0.986·37-s + 1.94·38-s − 1.26·40-s − 1.56·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 134 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 198 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 226 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.77149743030275547087844509666, −7.66026169664269645391456143371, −6.90205496839492655541880248691, −6.86351525870515771953629407719, −6.28661359246940961805771077587, −6.25365222743752711348058067337, −5.80595327823422032797086263217, −5.75210513860716419283317389935, −4.81697881762992402594272515941, −4.74024720856203321148457284825, −3.89057483439351344999634283754, −3.77847458107968562534567290533, −3.11955513324193006234013032071, −3.01288340437323589769415436169, −2.12024097263932706963544870876, −2.05047783562099962339180099616, −1.31392596698359125325415104831, −1.28115975618625175691611208527, 0, 0,
1.28115975618625175691611208527, 1.31392596698359125325415104831, 2.05047783562099962339180099616, 2.12024097263932706963544870876, 3.01288340437323589769415436169, 3.11955513324193006234013032071, 3.77847458107968562534567290533, 3.89057483439351344999634283754, 4.74024720856203321148457284825, 4.81697881762992402594272515941, 5.75210513860716419283317389935, 5.80595327823422032797086263217, 6.25365222743752711348058067337, 6.28661359246940961805771077587, 6.86351525870515771953629407719, 6.90205496839492655541880248691, 7.66026169664269645391456143371, 7.77149743030275547087844509666
