L(s) = 1 | − 2-s − 2·3-s − 2·4-s − 2·5-s + 2·6-s − 3·7-s + 3·8-s + 2·9-s + 2·10-s + 11-s + 4·12-s − 13-s + 3·14-s + 4·15-s + 16-s − 2·17-s − 2·18-s + 4·20-s + 6·21-s − 22-s + 2·23-s − 6·24-s − 5·25-s + 26-s − 6·27-s + 6·28-s − 3·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 1.06·8-s + 2/3·9-s + 0.632·10-s + 0.301·11-s + 1.15·12-s − 0.277·13-s + 0.801·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.894·20-s + 1.30·21-s − 0.213·22-s + 0.417·23-s − 1.22·24-s − 25-s + 0.196·26-s − 1.15·27-s + 1.13·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 149 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| 673 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 44 T + p T^{2} ) \) |
good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 75 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T - 89 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 146 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 35 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T - 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
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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
−14.5313283919, −13.8174585929, −13.5771270725, −13.1172928081, −12.6596356057, −12.2970673201, −11.8134139345, −11.2999508887, −11.1028266278, −10.3378325837, −10.0133509138, −9.53067473702, −9.16406414732, −8.76281687216, −8.04679040661, −7.71037425608, −6.98194853414, −6.70426036273, −5.94233602139, −5.44846515949, −4.91683594441, −4.07881694741, −3.91157222741, −3.04724986928, −1.62540141060, 0, 0,
1.62540141060, 3.04724986928, 3.91157222741, 4.07881694741, 4.91683594441, 5.44846515949, 5.94233602139, 6.70426036273, 6.98194853414, 7.71037425608, 8.04679040661, 8.76281687216, 9.16406414732, 9.53067473702, 10.0133509138, 10.3378325837, 11.1028266278, 11.2999508887, 11.8134139345, 12.2970673201, 12.6596356057, 13.1172928081, 13.5771270725, 13.8174585929, 14.5313283919