| L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s − 2·19-s − 2·23-s − 5·25-s + 2·31-s − 4·35-s + 6·37-s + 10·41-s − 4·43-s − 12·47-s + 3·49-s − 8·53-s − 4·55-s − 4·59-s − 8·67-s − 14·71-s − 12·73-s + 4·77-s + 4·79-s − 16·83-s − 6·89-s − 4·95-s − 8·97-s − 8·101-s + 2·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s − 0.458·19-s − 0.417·23-s − 25-s + 0.359·31-s − 0.676·35-s + 0.986·37-s + 1.56·41-s − 0.609·43-s − 1.75·47-s + 3/7·49-s − 1.09·53-s − 0.539·55-s − 0.520·59-s − 0.977·67-s − 1.66·71-s − 1.40·73-s + 0.455·77-s + 0.450·79-s − 1.75·83-s − 0.635·89-s − 0.410·95-s − 0.812·97-s − 0.796·101-s + 0.197·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 189 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 169 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 178 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.77839507185675824841724666184, −7.72909545769485688276484389422, −7.02008603442647904049580866077, −6.93426336504553864756297485507, −6.33850075272187957624060300598, −6.03592314694340509006234816501, −5.83597149907736961237338288896, −5.65089517907392505334987251137, −4.96809604809615829130272818313, −4.65810617440404148135119447060, −4.21400291355676059759753344580, −3.97007553192422330057244730909, −3.19096764472155789523079299833, −3.06859827210896982320436243239, −2.50623516113942303769594646027, −2.21363960816192593527625520787, −1.52145063621161452210954508431, −1.25758845434102922817243520612, 0, 0,
1.25758845434102922817243520612, 1.52145063621161452210954508431, 2.21363960816192593527625520787, 2.50623516113942303769594646027, 3.06859827210896982320436243239, 3.19096764472155789523079299833, 3.97007553192422330057244730909, 4.21400291355676059759753344580, 4.65810617440404148135119447060, 4.96809604809615829130272818313, 5.65089517907392505334987251137, 5.83597149907736961237338288896, 6.03592314694340509006234816501, 6.33850075272187957624060300598, 6.93426336504553864756297485507, 7.02008603442647904049580866077, 7.72909545769485688276484389422, 7.77839507185675824841724666184
