| L(s) = 1 | + 2·5-s − 4·7-s − 4·9-s − 4·11-s − 2·13-s − 4·17-s − 4·19-s + 3·25-s − 12·31-s − 8·35-s − 12·41-s + 8·43-s − 8·45-s + 4·47-s + 6·49-s − 12·53-s − 8·55-s − 12·59-s − 16·61-s + 16·63-s − 4·65-s + 4·67-s − 4·71-s + 16·77-s + 7·81-s + 12·83-s − 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 4/3·9-s − 1.20·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s + 3/5·25-s − 2.15·31-s − 1.35·35-s − 1.87·41-s + 1.21·43-s − 1.19·45-s + 0.583·47-s + 6/7·49-s − 1.64·53-s − 1.07·55-s − 1.56·59-s − 2.04·61-s + 2.01·63-s − 0.496·65-s + 0.488·67-s − 0.474·71-s + 1.82·77-s + 7/9·81-s + 1.31·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.513417429972597511157704400052, −9.380076667752319533641765511932, −8.944643708917373215364298707961, −8.749388772779863169025448597269, −7.87534144334087670257100206633, −7.84501417618635167893809023441, −7.07189073883090683697271068169, −6.68768278994150511040093646428, −6.17532441976322888621306830850, −6.04497850779007358243459993979, −5.38889417693049779700387133592, −5.13855855983570606187019462115, −4.52355324024085020876948882440, −3.82228375347913551862514226162, −3.02154194873371797986296120732, −3.00411203023157687500620411889, −2.26317203011178606909658408004, −1.79156297564052064721753575750, 0, 0,
1.79156297564052064721753575750, 2.26317203011178606909658408004, 3.00411203023157687500620411889, 3.02154194873371797986296120732, 3.82228375347913551862514226162, 4.52355324024085020876948882440, 5.13855855983570606187019462115, 5.38889417693049779700387133592, 6.04497850779007358243459993979, 6.17532441976322888621306830850, 6.68768278994150511040093646428, 7.07189073883090683697271068169, 7.84501417618635167893809023441, 7.87534144334087670257100206633, 8.749388772779863169025448597269, 8.944643708917373215364298707961, 9.380076667752319533641765511932, 9.513417429972597511157704400052
