| L(s) = 1 | + 3-s + 4-s − 2·7-s + 9-s − 5·11-s + 12-s + 16-s − 2·21-s + 2·25-s + 27-s − 2·28-s − 5·33-s + 36-s − 37-s − 9·41-s − 5·44-s + 2·47-s + 48-s − 7·49-s + 2·53-s − 2·63-s + 64-s − 4·67-s − 9·71-s + 10·73-s + 2·75-s + 10·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 1.50·11-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 − 0.870·33-s + 1/6·36-s − 0.164·37-s − 1.40·41-s − 0.753·44-s + 0.291·47-s + 0.144·48-s − 49-s + 0.274·53-s − 0.251·63-s + 1/8·64-s − 0.488·67-s − 1.06·71-s + 1.17·73-s + 0.230·75-s + 1.13·77-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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 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$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 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
−8.320013013357955201005568896505, −7.906304917693391098365879441046, −7.50037322635817695998657938426, −6.99689590419589900501941447443, −6.60659453908028561583562117040, −6.12061220736335014016127662535, −5.49578668719950258456885245464, −5.08362170238176331977905596489, −4.54452302452415974694591572569, −3.74782003463553909184363884855, −3.26678744520512449264461442651, −2.75411880047870456808843993369, −2.28335784710737921704798683269, −1.40285595218231090092364541261, 0,
1.40285595218231090092364541261, 2.28335784710737921704798683269, 2.75411880047870456808843993369, 3.26678744520512449264461442651, 3.74782003463553909184363884855, 4.54452302452415974694591572569, 5.08362170238176331977905596489, 5.49578668719950258456885245464, 6.12061220736335014016127662535, 6.60659453908028561583562117040, 6.99689590419589900501941447443, 7.50037322635817695998657938426, 7.906304917693391098365879441046, 8.320013013357955201005568896505
