| L(s) = 1 | − 2·5-s + 2·7-s − 4·11-s − 8·13-s + 6·17-s + 2·19-s + 12·23-s − 6·29-s − 8·31-s − 4·35-s − 4·41-s − 4·43-s − 10·47-s + 3·49-s + 2·53-s + 8·55-s − 8·59-s + 16·61-s + 16·65-s + 8·67-s − 6·71-s − 8·73-s − 8·77-s + 12·79-s − 18·83-s − 12·85-s − 12·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.20·11-s − 2.21·13-s + 1.45·17-s + 0.458·19-s + 2.50·23-s − 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.624·41-s − 0.609·43-s − 1.45·47-s + 3/7·49-s + 0.274·53-s + 1.07·55-s − 1.04·59-s + 2.04·61-s + 1.98·65-s + 0.977·67-s − 0.712·71-s − 0.936·73-s − 0.911·77-s + 1.35·79-s − 1.97·83-s − 1.30·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 36 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 60 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 112 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 144 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.41273172455490866247135357414, −7.31421030069009047839379644265, −7.04173865384601916746609387992, −6.81966215007009031488342702908, −5.92219760382809213557430117564, −5.78818050625183393982499936125, −5.16583143215371451030444266880, −5.05286515177346034143044927329, −4.98057801195994588005802727173, −4.61321328386928231412970215372, −3.86136669335360337338834127199, −3.65976875108184288472763538888, −3.07352470371142857286568009715, −3.02901787762875279513982962939, −2.35443053532807852030180187365, −2.08926892646436461320551620342, −1.39919491925572497014274298275, −0.983061595614113974541227485583, 0, 0,
0.983061595614113974541227485583, 1.39919491925572497014274298275, 2.08926892646436461320551620342, 2.35443053532807852030180187365, 3.02901787762875279513982962939, 3.07352470371142857286568009715, 3.65976875108184288472763538888, 3.86136669335360337338834127199, 4.61321328386928231412970215372, 4.98057801195994588005802727173, 5.05286515177346034143044927329, 5.16583143215371451030444266880, 5.78818050625183393982499936125, 5.92219760382809213557430117564, 6.81966215007009031488342702908, 7.04173865384601916746609387992, 7.31421030069009047839379644265, 7.41273172455490866247135357414
