| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 9-s + 2·10-s − 2·13-s + 16-s + 3·17-s − 18-s − 2·20-s − 25-s + 2·26-s − 11·29-s − 32-s − 3·34-s + 36-s + 7·37-s + 2·40-s − 2·41-s − 2·45-s + 49-s + 50-s − 2·52-s + 2·53-s + 11·58-s − 9·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.554·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.447·20-s − 1/5·25-s + 0.392·26-s − 2.04·29-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 1.15·37-s + 0.316·40-s − 0.312·41-s − 0.298·45-s + 1/7·49-s + 0.141·50-s − 0.277·52-s + 0.274·53-s + 1.44·58-s − 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 157600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 157600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 197 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 15 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.081354242011937318916867605422, −8.474033604486328838477276867625, −7.907047202967900160710107789563, −7.68599753240436163569675130597, −7.24686901475225602386396050434, −6.79420647032414861738451058375, −6.02510512592976249318245404623, −5.60548402588121341840880818922, −4.94667704943473045360302303731, −4.19656177361373599440201295365, −3.75918899058795346450437788877, −3.05790280413881408038332757046, −2.25317987835290883086068630557, −1.33104358846386158134372759561, 0,
1.33104358846386158134372759561, 2.25317987835290883086068630557, 3.05790280413881408038332757046, 3.75918899058795346450437788877, 4.19656177361373599440201295365, 4.94667704943473045360302303731, 5.60548402588121341840880818922, 6.02510512592976249318245404623, 6.79420647032414861738451058375, 7.24686901475225602386396050434, 7.68599753240436163569675130597, 7.907047202967900160710107789563, 8.474033604486328838477276867625, 9.081354242011937318916867605422
