| L(s) = 1 | − 3·3-s + 6·9-s − 5·11-s − 7·19-s + 25-s − 9·27-s + 15·33-s − 11·41-s − 2·43-s − 10·49-s + 21·57-s + 59-s − 16·67-s + 9·73-s − 3·75-s + 9·81-s − 3·83-s − 13·89-s + 2·97-s − 30·99-s − 13·107-s − 10·113-s − 121-s + 33·123-s + 127-s + 6·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 1.50·11-s − 1.60·19-s + 1/5·25-s − 1.73·27-s + 2.61·33-s − 1.71·41-s − 0.304·43-s − 1.42·49-s + 2.78·57-s + 0.130·59-s − 1.95·67-s + 1.05·73-s − 0.346·75-s + 81-s − 0.329·83-s − 1.37·89-s + 0.203·97-s − 3.01·99-s − 1.25·107-s − 0.940·113-s − 0.0909·121-s + 2.97·123-s + 0.0887·127-s + 0.528·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−6.96748571191941665117874660568, −6.68004856664632365282737869024, −6.38793181041715272972051380069, −5.91649038636657881937285852706, −5.46334269722461697779450310574, −5.04273662910579573624445714183, −4.87168840484880303096961679508, −4.28121261990243824199159043648, −3.85916531056371460401610578505, −3.09988844689213672372525736709, −2.56332673414763506193409678108, −1.85731584386660975352190299470, −1.27037267470513427470770986588, 0, 0,
1.27037267470513427470770986588, 1.85731584386660975352190299470, 2.56332673414763506193409678108, 3.09988844689213672372525736709, 3.85916531056371460401610578505, 4.28121261990243824199159043648, 4.87168840484880303096961679508, 5.04273662910579573624445714183, 5.46334269722461697779450310574, 5.91649038636657881937285852706, 6.38793181041715272972051380069, 6.68004856664632365282737869024, 6.96748571191941665117874660568
