| L(s) = 1 | + 3-s − 4-s + 2·7-s + 9-s − 3·11-s − 12-s + 16-s + 2·21-s − 2·25-s + 27-s − 2·28-s − 3·33-s − 36-s − 37-s − 13·41-s + 3·44-s − 14·47-s + 48-s − 7·49-s + 2·53-s + 2·63-s − 64-s + 4·67-s − 7·71-s − 10·73-s − 2·75-s − 6·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1/4·16-s + 0.436·21-s − 2/5·25-s + 0.192·27-s − 0.377·28-s − 0.522·33-s − 1/6·36-s − 0.164·37-s − 2.03·41-s + 0.452·44-s − 2.04·47-s + 0.144·48-s − 49-s + 0.274·53-s + 0.251·63-s − 1/8·64-s + 0.488·67-s − 0.830·71-s − 1.17·73-s − 0.230·75-s − 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 37 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 109 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
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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
−8.220316120494889406389046686956, −8.108829119441733574098335875681, −7.65318540138488602511211355763, −6.96858216182867601696485690963, −6.70775797238058662452768158227, −5.90329597383620010040173165726, −5.46347637110047668592042310892, −4.83447744193489313368734896791, −4.70157733520770343158405900874, −3.93864245017437603249384915481, −3.31581839791787025439367503731, −2.87493043254454627851146097470, −1.97487405866122722919949778162, −1.46128681204988400269737508691, 0,
1.46128681204988400269737508691, 1.97487405866122722919949778162, 2.87493043254454627851146097470, 3.31581839791787025439367503731, 3.93864245017437603249384915481, 4.70157733520770343158405900874, 4.83447744193489313368734896791, 5.46347637110047668592042310892, 5.90329597383620010040173165726, 6.70775797238058662452768158227, 6.96858216182867601696485690963, 7.65318540138488602511211355763, 8.108829119441733574098335875681, 8.220316120494889406389046686956
