| L(s) = 1 | + 2·3-s + 2·5-s + 3·9-s − 2·11-s + 2·13-s + 4·15-s − 2·17-s + 4·19-s − 12·23-s − 4·25-s + 4·27-s − 10·29-s − 14·31-s − 4·33-s − 4·37-s + 4·39-s − 8·41-s − 10·43-s + 6·45-s − 4·47-s − 2·49-s − 4·51-s − 12·53-s − 4·55-s + 8·57-s − 4·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.485·17-s + 0.917·19-s − 2.50·23-s − 4/5·25-s + 0.769·27-s − 1.85·29-s − 2.51·31-s − 0.696·33-s − 0.657·37-s + 0.640·39-s − 1.24·41-s − 1.52·43-s + 0.894·45-s − 0.583·47-s − 2/7·49-s − 0.560·51-s − 1.64·53-s − 0.539·55-s + 1.05·57-s − 0.520·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 116 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 180 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.69597131791864015009362641767, −7.56950803674488182985566901014, −7.22241049868047777774886470892, −6.89357675409022916530495156171, −6.17840073585047549482521104897, −6.06368091552583036293926000273, −5.77456363181654365610518175832, −5.39629118391642939488247088111, −4.81568293075435705697504544261, −4.69026469930325998227201605605, −3.94713003019103775969752572938, −3.66599511753818760381254893294, −3.28278953456966509281594157748, −3.22266118836494041139261346312, −2.24906003724402399616722035615, −2.02229211029712255447480411190, −1.65962894036552175605388417816, −1.57676203018402228764740793214, 0, 0,
1.57676203018402228764740793214, 1.65962894036552175605388417816, 2.02229211029712255447480411190, 2.24906003724402399616722035615, 3.22266118836494041139261346312, 3.28278953456966509281594157748, 3.66599511753818760381254893294, 3.94713003019103775969752572938, 4.69026469930325998227201605605, 4.81568293075435705697504544261, 5.39629118391642939488247088111, 5.77456363181654365610518175832, 6.06368091552583036293926000273, 6.17840073585047549482521104897, 6.89357675409022916530495156171, 7.22241049868047777774886470892, 7.56950803674488182985566901014, 7.69597131791864015009362641767
