| L(s) = 1 | + 2·7-s − 4·11-s + 4·13-s − 8·17-s + 2·19-s − 4·23-s − 2·25-s + 4·29-s + 8·31-s − 12·37-s − 4·41-s − 8·43-s − 4·47-s + 3·49-s − 12·53-s − 8·59-s − 12·61-s − 16·71-s + 20·73-s − 8·77-s + 24·79-s + 20·83-s + 12·89-s + 8·91-s − 12·97-s + 8·101-s + 16·107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s + 1.10·13-s − 1.94·17-s + 0.458·19-s − 0.834·23-s − 2/5·25-s + 0.742·29-s + 1.43·31-s − 1.97·37-s − 0.624·41-s − 1.21·43-s − 0.583·47-s + 3/7·49-s − 1.64·53-s − 1.04·59-s − 1.53·61-s − 1.89·71-s + 2.34·73-s − 0.911·77-s + 2.70·79-s + 2.19·83-s + 1.27·89-s + 0.838·91-s − 1.21·97-s + 0.796·101-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 234 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.48935662020286926813297623600, −7.37769003499262221919137583300, −6.63994601206952979782139814488, −6.45381371552092290498554303597, −6.17452172159072746067291518206, −6.09447244759399126547395746060, −5.22094896792251978155152435691, −4.97886634915747501555036272943, −4.80014796253553452130743610688, −4.69512817192142540259275188361, −3.85182797698426555506211960264, −3.68809063160133001641568423848, −3.28973185598106911419952532302, −2.80176277333637582187718452936, −2.17917649493817097690997598026, −2.13511727920923505958470566935, −1.46968303789891711414095216111, −1.10540297389681392526699458541, 0, 0,
1.10540297389681392526699458541, 1.46968303789891711414095216111, 2.13511727920923505958470566935, 2.17917649493817097690997598026, 2.80176277333637582187718452936, 3.28973185598106911419952532302, 3.68809063160133001641568423848, 3.85182797698426555506211960264, 4.69512817192142540259275188361, 4.80014796253553452130743610688, 4.97886634915747501555036272943, 5.22094896792251978155152435691, 6.09447244759399126547395746060, 6.17452172159072746067291518206, 6.45381371552092290498554303597, 6.63994601206952979782139814488, 7.37769003499262221919137583300, 7.48935662020286926813297623600
