| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s − 4·11-s − 2·13-s − 4·15-s − 6·19-s + 4·21-s − 4·23-s − 2·25-s − 4·27-s − 2·31-s + 8·33-s − 4·35-s − 8·37-s + 4·39-s − 2·41-s − 4·43-s + 6·45-s + 4·47-s − 6·49-s + 8·53-s − 8·55-s + 12·57-s − 8·59-s − 16·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 1.20·11-s − 0.554·13-s − 1.03·15-s − 1.37·19-s + 0.872·21-s − 0.834·23-s − 2/5·25-s − 0.769·27-s − 0.359·31-s + 1.39·33-s − 0.676·35-s − 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.609·43-s + 0.894·45-s + 0.583·47-s − 6/7·49-s + 1.09·53-s − 1.07·55-s + 1.58·57-s − 1.04·59-s − 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_4$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−9.433145564637886026903057584221, −9.351705555426971137426416943951, −8.759874431948671734139567460138, −8.275798701814878111155644201629, −7.72832915348017176959791716160, −7.43425858912194942112343560581, −6.85762564308701428781618770273, −6.52275900808355859074240311884, −6.02272004970955032967360176030, −5.84678043219083801586685780798, −5.29396245278535767365914782195, −5.03228506978679632896220437989, −4.21578031527768778010874229372, −4.16066266332991655782669416607, −3.07633394664636135572320304998, −2.80243939547291842600376812437, −1.87962937083684183301061413624, −1.65256455659439457675396194219, 0, 0,
1.65256455659439457675396194219, 1.87962937083684183301061413624, 2.80243939547291842600376812437, 3.07633394664636135572320304998, 4.16066266332991655782669416607, 4.21578031527768778010874229372, 5.03228506978679632896220437989, 5.29396245278535767365914782195, 5.84678043219083801586685780798, 6.02272004970955032967360176030, 6.52275900808355859074240311884, 6.85762564308701428781618770273, 7.43425858912194942112343560581, 7.72832915348017176959791716160, 8.275798701814878111155644201629, 8.759874431948671734139567460138, 9.351705555426971137426416943951, 9.433145564637886026903057584221
