| L(s) = 1 | − 2·5-s − 7-s − 11-s − 6·13-s + 5·17-s + 3·19-s + 7·23-s + 3·25-s − 29-s − 13·31-s + 2·35-s − 7·37-s + 2·41-s + 5·43-s + 3·47-s + 5·49-s − 2·53-s + 2·55-s − 24·59-s + 7·61-s + 12·65-s + 21·67-s + 10·71-s + 73-s + 77-s + 10·79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 0.301·11-s − 1.66·13-s + 1.21·17-s + 0.688·19-s + 1.45·23-s + 3/5·25-s − 0.185·29-s − 2.33·31-s + 0.338·35-s − 1.15·37-s + 0.312·41-s + 0.762·43-s + 0.437·47-s + 5/7·49-s − 0.274·53-s + 0.269·55-s − 3.12·59-s + 0.896·61-s + 1.48·65-s + 2.56·67-s + 1.18·71-s + 0.117·73-s + 0.113·77-s + 1.12·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74649600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74649600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 13 T + 86 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 68 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 74 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 116 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 21 T + 226 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 128 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−7.44544608607841525584324645050, −7.23529398857721765074769850853, −7.17860360766751986277973272129, −6.71903617738600392254930014112, −6.33559823973391553870163876312, −5.63212458147934713010068462966, −5.38666965501774457259977921807, −5.32194634759008254622039328890, −4.89823146167077997706276239890, −4.45632421721062209567537231088, −3.85159479933911040948865235713, −3.79834323348458649856024219133, −3.12583522198040680441420703712, −3.07578304740026323447591396681, −2.45350956309914639437888457554, −2.14455908008135793811157364649, −1.30827508715278682023731748763, −1.03391247551526228911919062431, 0, 0,
1.03391247551526228911919062431, 1.30827508715278682023731748763, 2.14455908008135793811157364649, 2.45350956309914639437888457554, 3.07578304740026323447591396681, 3.12583522198040680441420703712, 3.79834323348458649856024219133, 3.85159479933911040948865235713, 4.45632421721062209567537231088, 4.89823146167077997706276239890, 5.32194634759008254622039328890, 5.38666965501774457259977921807, 5.63212458147934713010068462966, 6.33559823973391553870163876312, 6.71903617738600392254930014112, 7.17860360766751986277973272129, 7.23529398857721765074769850853, 7.44544608607841525584324645050
