| L(s) = 1 | − 3-s + 4-s − 7·7-s + 9-s − 12-s − 6·13-s + 16-s − 8·19-s + 7·21-s + 25-s − 27-s − 7·28-s − 4·31-s + 36-s − 3·37-s + 6·39-s + 10·43-s − 48-s + 25·49-s − 6·52-s + 8·57-s − 9·61-s − 7·63-s + 64-s − 17·67-s − 16·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 2.64·7-s + 1/3·9-s − 0.288·12-s − 1.66·13-s + 1/4·16-s − 1.83·19-s + 1.52·21-s + 1/5·25-s − 0.192·27-s − 1.32·28-s − 0.718·31-s + 1/6·36-s − 0.493·37-s + 0.960·39-s + 1.52·43-s − 0.144·48-s + 25/7·49-s − 0.832·52-s + 1.05·57-s − 1.15·61-s − 0.881·63-s + 1/8·64-s − 2.07·67-s − 1.87·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900828 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900828 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 7 T + p T^{2} ) \) |
| 439 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 15 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.44766168901559848281956214569, −7.16039140311490985086207437314, −6.88629089189577388605062239828, −6.34040149781041909463691393091, −6.06364136271471563576810247811, −5.72821577122165821821982650462, −5.08642881140822717163628046539, −4.29670057918536442031652442394, −4.13854604940762454721045351463, −3.31456892344396394200533806174, −2.75905728761456321443446741049, −2.53995835830879160028882798856, −1.56439592122180687572869765272, 0, 0,
1.56439592122180687572869765272, 2.53995835830879160028882798856, 2.75905728761456321443446741049, 3.31456892344396394200533806174, 4.13854604940762454721045351463, 4.29670057918536442031652442394, 5.08642881140822717163628046539, 5.72821577122165821821982650462, 6.06364136271471563576810247811, 6.34040149781041909463691393091, 6.88629089189577388605062239828, 7.16039140311490985086207437314, 7.44766168901559848281956214569
