| L(s) = 1 | − 2-s + 4-s − 6·7-s − 8-s − 2·9-s + 10·13-s + 6·14-s + 16-s − 8·17-s + 2·18-s + 4·19-s − 8·23-s − 6·25-s − 10·26-s − 6·28-s + 6·29-s − 16·31-s − 32-s + 8·34-s − 2·36-s + 12·37-s − 4·38-s − 12·43-s + 8·46-s − 2·47-s + 14·49-s + 6·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 2.26·7-s − 0.353·8-s − 2/3·9-s + 2.77·13-s + 1.60·14-s + 1/4·16-s − 1.94·17-s + 0.471·18-s + 0.917·19-s − 1.66·23-s − 6/5·25-s − 1.96·26-s − 1.13·28-s + 1.11·29-s − 2.87·31-s − 0.176·32-s + 1.37·34-s − 1/3·36-s + 1.97·37-s − 0.648·38-s − 1.82·43-s + 1.17·46-s − 0.291·47-s + 2·49-s + 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−15.7998429749, −15.5438761265, −14.8956300270, −14.0828582458, −13.5603503236, −13.3134015251, −13.0270778643, −12.3535797825, −11.5374061458, −11.4720141053, −10.5945207017, −10.5519389715, −9.48966114694, −9.33402764925, −9.03354891525, −8.06112795724, −8.02435745103, −6.70748925180, −6.57446374298, −6.00935780264, −5.69856539644, −4.19048615141, −3.54075497824, −3.11645057415, −1.87189238139, 0,
1.87189238139, 3.11645057415, 3.54075497824, 4.19048615141, 5.69856539644, 6.00935780264, 6.57446374298, 6.70748925180, 8.02435745103, 8.06112795724, 9.03354891525, 9.33402764925, 9.48966114694, 10.5519389715, 10.5945207017, 11.4720141053, 11.5374061458, 12.3535797825, 13.0270778643, 13.3134015251, 13.5603503236, 14.0828582458, 14.8956300270, 15.5438761265, 15.7998429749
