| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 4·11-s + 2·13-s + 4·14-s + 5·16-s + 10·17-s − 8·19-s − 8·22-s + 6·23-s − 4·26-s − 6·28-s + 2·29-s + 2·31-s − 6·32-s − 20·34-s + 12·37-s + 16·38-s − 8·41-s + 14·43-s + 12·44-s − 12·46-s − 4·47-s + 3·49-s + 6·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 5/4·16-s + 2.42·17-s − 1.83·19-s − 1.70·22-s + 1.25·23-s − 0.784·26-s − 1.13·28-s + 0.371·29-s + 0.359·31-s − 1.06·32-s − 3.42·34-s + 1.97·37-s + 2.59·38-s − 1.24·41-s + 2.13·43-s + 1.80·44-s − 1.76·46-s − 0.583·47-s + 3/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.009939047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009939047\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 18 T + 175 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 248 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 219 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 188 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.83731308506799140536824553872, −7.76741337362634618365228764842, −7.22354702466682185675937085424, −6.86415138530147717806438025455, −6.43415733256265357033133317026, −6.39857183140286740449115678180, −6.03911324460189041572831435014, −5.74449427499356053822802540462, −5.09861468538997760697350102213, −4.91984699178279064789370734755, −4.10640812115958885009445868085, −4.04168262799334491690291267136, −3.41798970470322404365022221665, −3.26989023278686784451804882221, −2.56442670435538539188394207030, −2.55483664549814423535565466541, −1.64012075372933921360216635189, −1.39387882222477001709234003344, −0.847183912741118160516919371180, −0.54190439858938777945377724963,
0.54190439858938777945377724963, 0.847183912741118160516919371180, 1.39387882222477001709234003344, 1.64012075372933921360216635189, 2.55483664549814423535565466541, 2.56442670435538539188394207030, 3.26989023278686784451804882221, 3.41798970470322404365022221665, 4.04168262799334491690291267136, 4.10640812115958885009445868085, 4.91984699178279064789370734755, 5.09861468538997760697350102213, 5.74449427499356053822802540462, 6.03911324460189041572831435014, 6.39857183140286740449115678180, 6.43415733256265357033133317026, 6.86415138530147717806438025455, 7.22354702466682185675937085424, 7.76741337362634618365228764842, 7.83731308506799140536824553872
