| L(s) = 1 | − 2-s + 4-s − 6·7-s − 8-s − 4·11-s − 10·13-s + 6·14-s + 16-s − 6·17-s + 2·19-s + 4·22-s − 14·23-s − 25-s + 10·26-s − 6·28-s + 8·29-s − 2·31-s − 32-s + 6·34-s + 10·37-s − 2·38-s − 10·41-s − 12·43-s − 4·44-s + 14·46-s + 12·47-s + 17·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 2.26·7-s − 0.353·8-s − 1.20·11-s − 2.77·13-s + 1.60·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s + 0.852·22-s − 2.91·23-s − 1/5·25-s + 1.96·26-s − 1.13·28-s + 1.48·29-s − 0.359·31-s − 0.176·32-s + 1.02·34-s + 1.64·37-s − 0.324·38-s − 1.56·41-s − 1.82·43-s − 0.603·44-s + 2.06·46-s + 1.75·47-s + 17/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−13.9771172460, −13.1830201225, −13.0764853816, −12.6928441524, −12.0752752420, −11.9118486001, −11.5182641363, −10.6145223108, −10.1152647782, −9.89963732804, −9.87204525884, −9.41597276453, −8.57076096668, −8.26804890939, −7.70214011733, −7.15250974800, −6.73860900675, −6.52094529535, −5.66605061214, −5.37363015538, −4.48730897908, −3.97860684060, −3.05104257304, −2.44850978333, −2.31390719923, 0, 0,
2.31390719923, 2.44850978333, 3.05104257304, 3.97860684060, 4.48730897908, 5.37363015538, 5.66605061214, 6.52094529535, 6.73860900675, 7.15250974800, 7.70214011733, 8.26804890939, 8.57076096668, 9.41597276453, 9.87204525884, 9.89963732804, 10.1152647782, 10.6145223108, 11.5182641363, 11.9118486001, 12.0752752420, 12.6928441524, 13.0764853816, 13.1830201225, 13.9771172460
