| L(s) = 1 | − 2·4-s − 2·7-s + 9-s − 2·11-s + 4·16-s − 2·23-s − 25-s + 4·28-s + 4·29-s − 2·36-s + 2·37-s − 14·43-s + 4·44-s − 3·49-s + 8·53-s − 2·63-s − 8·64-s − 8·67-s − 24·71-s + 4·77-s − 8·79-s − 8·81-s + 4·92-s − 2·99-s + 2·100-s − 16·107-s − 24·109-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 16-s − 0.417·23-s − 1/5·25-s + 0.755·28-s + 0.742·29-s − 1/3·36-s + 0.328·37-s − 2.13·43-s + 0.603·44-s − 3/7·49-s + 1.09·53-s − 0.251·63-s − 64-s − 0.977·67-s − 2.84·71-s + 0.455·77-s − 0.900·79-s − 8/9·81-s + 0.417·92-s − 0.201·99-s + 1/5·100-s − 1.54·107-s − 2.29·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
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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
−9.530726762458101409845708973625, −9.070142372419595448498611775310, −8.438386145726038831072500723921, −8.186020340391007454180225750519, −7.51110857423172456077324131289, −6.97164380745272621726069145499, −6.37791787319586287423789369386, −5.75207414829334705304948116898, −5.29181140436576152015639453623, −4.54337057823815951116088222595, −4.15230772973903661454488682421, −3.31219162858747036612992275826, −2.78529002426946111321512338978, −1.50248106348787892601161790396, 0,
1.50248106348787892601161790396, 2.78529002426946111321512338978, 3.31219162858747036612992275826, 4.15230772973903661454488682421, 4.54337057823815951116088222595, 5.29181140436576152015639453623, 5.75207414829334705304948116898, 6.37791787319586287423789369386, 6.97164380745272621726069145499, 7.51110857423172456077324131289, 8.186020340391007454180225750519, 8.438386145726038831072500723921, 9.070142372419595448498611775310, 9.530726762458101409845708973625
