| L(s) = 1 | − 4·3-s − 4·5-s − 4·7-s + 7·9-s − 4·11-s − 13-s + 16·15-s − 17-s − 8·19-s + 16·21-s − 23-s + 3·25-s − 4·27-s − 29-s + 31-s + 16·33-s + 16·35-s + 7·37-s + 4·39-s + 5·41-s − 6·43-s − 28·45-s + 4·47-s + 3·49-s + 4·51-s − 15·53-s + 16·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s − 1.51·7-s + 7/3·9-s − 1.20·11-s − 0.277·13-s + 4.13·15-s − 0.242·17-s − 1.83·19-s + 3.49·21-s − 0.208·23-s + 3/5·25-s − 0.769·27-s − 0.185·29-s + 0.179·31-s + 2.78·33-s + 2.70·35-s + 1.15·37-s + 0.640·39-s + 0.780·41-s − 0.914·43-s − 4.17·45-s + 0.583·47-s + 3/7·49-s + 0.560·51-s − 2.06·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 691 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 38 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 49 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 40 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 121 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−16.0500424610, −15.8351235472, −15.5101957298, −14.9897715596, −14.3698371963, −13.2484281159, −13.1290529901, −12.4975392187, −12.2099396995, −11.8701716449, −11.1957820433, −10.8377837457, −10.7192700793, −9.85989513158, −9.44793568370, −8.38393961412, −7.97846575436, −7.35390365950, −6.65324845722, −6.20893369004, −5.84171526201, −4.94013524184, −4.43903540517, −3.73618041581, −2.73130520401, 0, 0,
2.73130520401, 3.73618041581, 4.43903540517, 4.94013524184, 5.84171526201, 6.20893369004, 6.65324845722, 7.35390365950, 7.97846575436, 8.38393961412, 9.44793568370, 9.85989513158, 10.7192700793, 10.8377837457, 11.1957820433, 11.8701716449, 12.2099396995, 12.4975392187, 13.1290529901, 13.2484281159, 14.3698371963, 14.9897715596, 15.5101957298, 15.8351235472, 16.0500424610
