| L(s) = 1 | − 3-s − 2·5-s − 2·11-s − 8·13-s + 2·15-s − 6·17-s − 8·19-s + 6·23-s + 5·25-s + 27-s − 20·29-s − 4·31-s + 2·33-s − 6·37-s + 8·39-s − 12·41-s + 8·43-s − 8·47-s + 6·51-s − 2·53-s + 4·55-s + 8·57-s + 4·59-s + 8·61-s + 16·65-s + 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.603·11-s − 2.21·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s + 1.25·23-s + 25-s + 0.192·27-s − 3.71·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s + 1.28·39-s − 1.87·41-s + 1.21·43-s − 1.16·47-s + 0.840·51-s − 0.274·53-s + 0.539·55-s + 1.05·57-s + 0.520·59-s + 1.02·61-s + 1.98·65-s + 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−10.44762962121687293381199143183, −10.36758497730667856935644977680, −9.410304692521820771554835525408, −9.252782538482654723176739408920, −8.765490601545603914125011455773, −8.290665314179061344772246094631, −7.58623274027051951656007176582, −7.41671156228122135649852881403, −6.79172440819309870546315464275, −6.73046060607173898539584677331, −5.74190479357973473147643879227, −5.34430176953819470610031684652, −4.71287893562689173074985731068, −4.63821733801954034466714438835, −3.75178103956797459541164325423, −3.28601497984755703972909028945, −2.16795692825738455667006861215, −2.10996305030728947539194238780, 0, 0,
2.10996305030728947539194238780, 2.16795692825738455667006861215, 3.28601497984755703972909028945, 3.75178103956797459541164325423, 4.63821733801954034466714438835, 4.71287893562689173074985731068, 5.34430176953819470610031684652, 5.74190479357973473147643879227, 6.73046060607173898539584677331, 6.79172440819309870546315464275, 7.41671156228122135649852881403, 7.58623274027051951656007176582, 8.290665314179061344772246094631, 8.765490601545603914125011455773, 9.252782538482654723176739408920, 9.410304692521820771554835525408, 10.36758497730667856935644977680, 10.44762962121687293381199143183
