| L(s) = 1 | − 2·3-s + 2·7-s + 9-s − 4·13-s − 4·19-s − 4·21-s + 6·25-s + 4·27-s + 8·31-s − 12·37-s + 8·39-s − 4·43-s + 3·49-s + 8·57-s − 4·61-s + 2·63-s + 4·67-s − 4·73-s − 12·75-s + 8·79-s − 11·81-s − 8·91-s − 16·93-s − 20·97-s + 8·103-s + 4·109-s + 24·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.917·19-s − 0.872·21-s + 6/5·25-s + 0.769·27-s + 1.43·31-s − 1.97·37-s + 1.28·39-s − 0.609·43-s + 3/7·49-s + 1.05·57-s − 0.512·61-s + 0.251·63-s + 0.488·67-s − 0.468·73-s − 1.38·75-s + 0.900·79-s − 1.22·81-s − 0.838·91-s − 1.65·93-s − 2.03·97-s + 0.788·103-s + 0.383·109-s + 2.27·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.330116886932601228925999241879, −7.994815217852826414720828233198, −7.23248402340205053458264005937, −6.93433509661449716195569856496, −6.51693735340967229182204042100, −6.04543017384416069491086362475, −5.35140007878757908805997429070, −5.09933647284904066642737058868, −4.65148783837025373838027577934, −4.25039703125381890307800361330, −3.34400886547571178943348903873, −2.69196473366569150795116406952, −2.00140362676266510508581492466, −1.10761117265302976138172283253, 0,
1.10761117265302976138172283253, 2.00140362676266510508581492466, 2.69196473366569150795116406952, 3.34400886547571178943348903873, 4.25039703125381890307800361330, 4.65148783837025373838027577934, 5.09933647284904066642737058868, 5.35140007878757908805997429070, 6.04543017384416069491086362475, 6.51693735340967229182204042100, 6.93433509661449716195569856496, 7.23248402340205053458264005937, 7.994815217852826414720828233198, 8.330116886932601228925999241879
