| L(s) = 1 | − 2·2-s + 3·4-s − 4·7-s − 4·8-s − 6·9-s + 4·11-s + 8·14-s + 5·16-s + 12·18-s − 8·22-s + 2·23-s + 6·25-s − 12·28-s + 4·29-s − 6·32-s − 18·36-s − 8·37-s + 20·43-s + 12·44-s − 4·46-s + 9·49-s − 12·50-s − 8·53-s + 16·56-s − 8·58-s + 24·63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s − 2·9-s + 1.20·11-s + 2.13·14-s + 5/4·16-s + 2.82·18-s − 1.70·22-s + 0.417·23-s + 6/5·25-s − 2.26·28-s + 0.742·29-s − 1.06·32-s − 3·36-s − 1.31·37-s + 3.04·43-s + 1.80·44-s − 0.589·46-s + 9/7·49-s − 1.69·50-s − 1.09·53-s + 2.13·56-s − 1.05·58-s + 3.02·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.052261676116547781964364742438, −8.971935452602447588346290189052, −8.659904731263005672439155026721, −7.927191639260373557159137642919, −7.36385009266802735585262564707, −6.69441913641732873591204403077, −6.52483267466185238810930678826, −5.84432417959727908710252168629, −5.57653141631531896664928182342, −4.45284615884695323748315797805, −3.53555410864630123788166205174, −2.91261745614448351218691160624, −2.63232737474552247848840602025, −1.20032515009423146216969398573, 0,
1.20032515009423146216969398573, 2.63232737474552247848840602025, 2.91261745614448351218691160624, 3.53555410864630123788166205174, 4.45284615884695323748315797805, 5.57653141631531896664928182342, 5.84432417959727908710252168629, 6.52483267466185238810930678826, 6.69441913641732873591204403077, 7.36385009266802735585262564707, 7.927191639260373557159137642919, 8.659904731263005672439155026721, 8.971935452602447588346290189052, 9.052261676116547781964364742438
