| L(s) = 1 | + 2-s − 2·3-s + 4-s − 3·5-s − 2·6-s − 4·7-s + 8-s − 3·10-s − 4·11-s − 2·12-s + 4·13-s − 4·14-s + 6·15-s + 16-s − 2·17-s − 2·19-s − 3·20-s + 8·21-s − 4·22-s − 4·23-s − 2·24-s + 8·25-s + 4·26-s + 2·27-s − 4·28-s + 6·30-s − 31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 1.51·7-s + 0.353·8-s − 0.948·10-s − 1.20·11-s − 0.577·12-s + 1.10·13-s − 1.06·14-s + 1.54·15-s + 1/4·16-s − 0.485·17-s − 0.458·19-s − 0.670·20-s + 1.74·21-s − 0.852·22-s − 0.834·23-s − 0.408·24-s + 8/5·25-s + 0.784·26-s + 0.384·27-s − 0.755·28-s + 1.09·30-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 251 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 18 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T - 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 120 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 86 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.5679335995, −16.2176342440, −15.8706017004, −15.5025533089, −15.1980300008, −14.3035677855, −13.7467188774, −13.0978075328, −12.8107733207, −12.3390054607, −11.8130169015, −11.3094733827, −10.7883317879, −10.5583546423, −9.74654623717, −8.88472056867, −8.25528534083, −7.64167672002, −6.85923579946, −6.33926390154, −5.83138217656, −5.14333894993, −4.23615075130, −3.58077840272, −2.77329145704, 0,
2.77329145704, 3.58077840272, 4.23615075130, 5.14333894993, 5.83138217656, 6.33926390154, 6.85923579946, 7.64167672002, 8.25528534083, 8.88472056867, 9.74654623717, 10.5583546423, 10.7883317879, 11.3094733827, 11.8130169015, 12.3390054607, 12.8107733207, 13.0978075328, 13.7467188774, 14.3035677855, 15.1980300008, 15.5025533089, 15.8706017004, 16.2176342440, 16.5679335995
