| L(s) = 1 | − 3-s + 2·5-s − 7-s − 4·9-s − 11-s − 3·13-s − 2·15-s + 17-s + 3·19-s + 21-s − 2·23-s + 3·25-s + 6·27-s − 14·29-s − 7·31-s + 33-s − 2·35-s + 4·37-s + 3·39-s − 9·41-s − 8·45-s − 6·47-s − 12·49-s − 51-s − 8·53-s − 2·55-s − 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s − 4/3·9-s − 0.301·11-s − 0.832·13-s − 0.516·15-s + 0.242·17-s + 0.688·19-s + 0.218·21-s − 0.417·23-s + 3/5·25-s + 1.15·27-s − 2.59·29-s − 1.25·31-s + 0.174·33-s − 0.338·35-s + 0.657·37-s + 0.480·39-s − 1.40·41-s − 1.19·45-s − 0.875·47-s − 1.71·49-s − 0.140·51-s − 1.09·53-s − 0.269·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 27 T + 375 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
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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
−9.192606943682537249560077900761, −8.699434749526360806222975886583, −8.337931721751252323402596797343, −7.82777677525183601533590119977, −7.30237624592909503089433211277, −7.27970698667016167621697858886, −6.38680272369479985109405182628, −6.29585942859912162290880847594, −5.67981252570599653848758504594, −5.57439464092130833527045222897, −5.03172832639205616633759214658, −4.89146941254222277346751527699, −3.96757502714772256063808846325, −3.52668189144133905567600257059, −2.91493048445896874514802456851, −2.71591056579124554536689308591, −1.81529603433925383725961295886, −1.53776077713781743186760966496, 0, 0,
1.53776077713781743186760966496, 1.81529603433925383725961295886, 2.71591056579124554536689308591, 2.91493048445896874514802456851, 3.52668189144133905567600257059, 3.96757502714772256063808846325, 4.89146941254222277346751527699, 5.03172832639205616633759214658, 5.57439464092130833527045222897, 5.67981252570599653848758504594, 6.29585942859912162290880847594, 6.38680272369479985109405182628, 7.27970698667016167621697858886, 7.30237624592909503089433211277, 7.82777677525183601533590119977, 8.337931721751252323402596797343, 8.699434749526360806222975886583, 9.192606943682537249560077900761
