| L(s) = 1 | − 2·2-s + 2·3-s + 4-s + 6·5-s − 4·6-s − 2·7-s + 2·8-s + 9-s − 12·10-s + 2·11-s + 2·12-s + 4·14-s + 12·15-s − 4·16-s − 2·18-s − 2·19-s + 6·20-s − 4·21-s − 4·22-s − 4·23-s + 4·24-s + 17·25-s − 2·27-s − 2·28-s + 20·29-s − 24·30-s + 10·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 2.68·5-s − 1.63·6-s − 0.755·7-s + 0.707·8-s + 1/3·9-s − 3.79·10-s + 0.603·11-s + 0.577·12-s + 1.06·14-s + 3.09·15-s − 16-s − 0.471·18-s − 0.458·19-s + 1.34·20-s − 0.872·21-s − 0.852·22-s − 0.834·23-s + 0.816·24-s + 17/5·25-s − 0.384·27-s − 0.377·28-s + 3.71·29-s − 4.38·30-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.128842440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.128842440\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 32 T^{2} + 8 T^{3} + 1407 T^{4} + 8 p T^{5} - 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 10 T + 45 T^{2} + 70 T^{3} - 1036 T^{4} + 70 p T^{5} + 45 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T - 8 T^{2} + 16 T^{3} + 1447 T^{4} + 16 p T^{5} - 8 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 22 T^{2} + 336 T^{3} + 5907 T^{4} + 336 p T^{5} + 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2 T - 95 T^{2} + 14 T^{3} + 6780 T^{4} + 14 p T^{5} - 95 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T - 65 T^{2} + 98 T^{3} + 1044 T^{4} + 98 p T^{5} - 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T - 102 T^{2} + 16 T^{3} + 9227 T^{4} + 16 p T^{5} - 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 294 T^{2} - 3264 T^{3} + 32147 T^{4} - 3264 p T^{5} + 294 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 123 T^{2} - 6 T^{3} + 16196 T^{4} - 6 p T^{5} - 123 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 16 T + 32 T^{2} + 736 T^{3} + 19471 T^{4} + 736 p T^{5} + 32 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 14 T + 241 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.49667934836725759437598019541, −7.00271614096463559887752895961, −6.78004144603795916847284650170, −6.77167011765782445392694714894, −6.57089355671868198704372718759, −6.27734324818097041767077128382, −6.10600809800832053534029359640, −5.92300055147662346052161628481, −5.55812128442306022070008673540, −5.36796568687533943277449717096, −5.03098342559863342097605207453, −4.80149785452545287022380112876, −4.31731730877847880282540004773, −4.24211277218324257306097834592, −4.20097969927766368715433452917, −3.38285843419802107568911980762, −3.16270185061131601814143916962, −3.13088137543958099007457933335, −2.48752745332071949595791546200, −2.43503362928647196856039301507, −2.19838035359858025859853378702, −1.85416925279357554130139184771, −1.41296649158129671546084947271, −0.894268635537305127880504935690, −0.75892320862623553526495040214,
0.75892320862623553526495040214, 0.894268635537305127880504935690, 1.41296649158129671546084947271, 1.85416925279357554130139184771, 2.19838035359858025859853378702, 2.43503362928647196856039301507, 2.48752745332071949595791546200, 3.13088137543958099007457933335, 3.16270185061131601814143916962, 3.38285843419802107568911980762, 4.20097969927766368715433452917, 4.24211277218324257306097834592, 4.31731730877847880282540004773, 4.80149785452545287022380112876, 5.03098342559863342097605207453, 5.36796568687533943277449717096, 5.55812128442306022070008673540, 5.92300055147662346052161628481, 6.10600809800832053534029359640, 6.27734324818097041767077128382, 6.57089355671868198704372718759, 6.77167011765782445392694714894, 6.78004144603795916847284650170, 7.00271614096463559887752895961, 7.49667934836725759437598019541
