| L(s) = 1 | + 2·3-s − 2·5-s − 4·7-s + 3·9-s + 8·13-s − 4·15-s − 8·17-s − 2·19-s − 8·21-s − 4·23-s + 3·25-s + 4·27-s − 4·29-s + 4·31-s + 8·35-s + 8·37-s + 16·39-s − 12·41-s − 4·43-s − 6·45-s − 12·47-s − 16·51-s + 8·53-s − 4·57-s − 12·63-s − 16·65-s − 24·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1.51·7-s + 9-s + 2.21·13-s − 1.03·15-s − 1.94·17-s − 0.458·19-s − 1.74·21-s − 0.834·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s + 0.718·31-s + 1.35·35-s + 1.31·37-s + 2.56·39-s − 1.87·41-s − 0.609·43-s − 0.894·45-s − 1.75·47-s − 2.24·51-s + 1.09·53-s − 0.529·57-s − 1.51·63-s − 1.98·65-s − 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 88 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−8.241015763213273801907178317262, −7.912672064533223106069533593596, −7.52255334983447452150479148029, −6.93734567889786793596971120676, −6.74354466956762820580675553748, −6.42369365317686094962313402552, −6.02866466030532208362763017489, −5.93716324673394462646985309396, −4.91565969754336242605934988608, −4.72014923432151107362661793016, −4.15065487325445451763889940085, −3.80542225875426960661118457347, −3.65222852485894003003971265276, −3.22116822404464695522473897130, −2.71945909697686855898127100776, −2.42347099106709586935239125445, −1.51898054597079972778560392322, −1.39003105284735473389818470434, 0, 0,
1.39003105284735473389818470434, 1.51898054597079972778560392322, 2.42347099106709586935239125445, 2.71945909697686855898127100776, 3.22116822404464695522473897130, 3.65222852485894003003971265276, 3.80542225875426960661118457347, 4.15065487325445451763889940085, 4.72014923432151107362661793016, 4.91565969754336242605934988608, 5.93716324673394462646985309396, 6.02866466030532208362763017489, 6.42369365317686094962313402552, 6.74354466956762820580675553748, 6.93734567889786793596971120676, 7.52255334983447452150479148029, 7.912672064533223106069533593596, 8.241015763213273801907178317262
