| L(s) = 1 | − 2-s + 4-s − 2·5-s − 5·7-s − 3·8-s + 2·10-s − 11-s − 2·13-s + 5·14-s + 16-s + 17-s − 2·19-s − 2·20-s + 22-s − 9·23-s + 3·25-s + 2·26-s − 5·28-s + 6·29-s + 12·31-s + 32-s − 34-s + 10·35-s − 9·37-s + 2·38-s + 6·40-s − 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.88·7-s − 1.06·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 1.33·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s + 0.213·22-s − 1.87·23-s + 3/5·25-s + 0.392·26-s − 0.944·28-s + 1.11·29-s + 2.15·31-s + 0.176·32-s − 0.171·34-s + 1.69·35-s − 1.47·37-s + 0.324·38-s + 0.948·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 25 T + 294 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 122 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 160 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−10.25647380054202828006587558414, −10.04323222908775155559777351237, −9.693664612531849309713983016073, −9.279297698324351119466275768764, −8.536522635467047420772977550802, −8.460648932664054176160298451417, −7.925785096726287958543361634388, −7.38527339122787260506101003182, −6.83819828586910433106762194099, −6.33191804757917848346117094712, −6.30593074763065902148610787542, −5.64656296920201328406111053153, −4.54113818883496080611787069092, −4.51937609989202770452600308295, −3.46360417048540056927518683779, −2.99613946156065026333088922268, −2.82342396833124201444253741024, −1.66120589734537228912904795573, 0, 0,
1.66120589734537228912904795573, 2.82342396833124201444253741024, 2.99613946156065026333088922268, 3.46360417048540056927518683779, 4.51937609989202770452600308295, 4.54113818883496080611787069092, 5.64656296920201328406111053153, 6.30593074763065902148610787542, 6.33191804757917848346117094712, 6.83819828586910433106762194099, 7.38527339122787260506101003182, 7.925785096726287958543361634388, 8.460648932664054176160298451417, 8.536522635467047420772977550802, 9.279297698324351119466275768764, 9.693664612531849309713983016073, 10.04323222908775155559777351237, 10.25647380054202828006587558414
