| L(s) = 1 | − 7-s − 5·9-s − 10·11-s − 12·23-s + 25-s − 18·29-s + 12·37-s + 12·43-s + 49-s − 24·53-s + 5·63-s − 8·67-s + 16·71-s + 10·77-s − 6·79-s + 16·81-s + 50·99-s + 4·107-s − 22·109-s − 12·113-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 5/3·9-s − 3.01·11-s − 2.50·23-s + 1/5·25-s − 3.34·29-s + 1.97·37-s + 1.82·43-s + 1/7·49-s − 3.29·53-s + 0.629·63-s − 0.977·67-s + 1.89·71-s + 1.13·77-s − 0.675·79-s + 16/9·81-s + 5.02·99-s + 0.386·107-s − 2.10·109-s − 1.12·113-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 548800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.74465662186675135601071019089, −7.69912855129361241271554591687, −7.61068895852562797070369378134, −6.28715004025144947864257791694, −6.17106315121314565062659046983, −5.55365867270428984617540263559, −5.45734437575009036836059962399, −4.85287109801086584074848656984, −4.06182673812785905054132006247, −3.56542759922310316766610018532, −2.71647306061986597321844269058, −2.60467600555756229149551481858, −1.93520907898928704565581532153, 0, 0,
1.93520907898928704565581532153, 2.60467600555756229149551481858, 2.71647306061986597321844269058, 3.56542759922310316766610018532, 4.06182673812785905054132006247, 4.85287109801086584074848656984, 5.45734437575009036836059962399, 5.55365867270428984617540263559, 6.17106315121314565062659046983, 6.28715004025144947864257791694, 7.61068895852562797070369378134, 7.69912855129361241271554591687, 7.74465662186675135601071019089
