| L(s) = 1 | − 2·3-s + 9-s − 2·11-s − 6·23-s − 5·25-s + 4·27-s − 4·31-s + 4·33-s − 18·37-s + 2·47-s − 12·49-s − 6·53-s − 16·59-s − 14·67-s + 12·69-s − 24·71-s + 10·75-s − 11·81-s − 4·89-s + 8·93-s + 10·97-s − 2·99-s − 2·103-s + 36·111-s + 6·113-s − 7·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.603·11-s − 1.25·23-s − 25-s + 0.769·27-s − 0.718·31-s + 0.696·33-s − 2.95·37-s + 0.291·47-s − 1.71·49-s − 0.824·53-s − 2.08·59-s − 1.71·67-s + 1.44·69-s − 2.84·71-s + 1.15·75-s − 1.22·81-s − 0.423·89-s + 0.829·93-s + 1.01·97-s − 0.201·99-s − 0.197·103-s + 3.41·111-s + 0.564·113-s − 0.636·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.36963711644089518142039272150, −6.97653088727263142756389606857, −6.37661064122731610749270783409, −6.04798848203660821052489182951, −5.72338911596447562569505218039, −5.32810490560116582990412818264, −4.70230174869687218784713714020, −4.56807903011406740199498793887, −3.76862795660492913201215671552, −3.28560629440378329367359919746, −2.78803787834219559133058057267, −1.77006532744428655728355255651, −1.60091467037784679419696293720, 0, 0,
1.60091467037784679419696293720, 1.77006532744428655728355255651, 2.78803787834219559133058057267, 3.28560629440378329367359919746, 3.76862795660492913201215671552, 4.56807903011406740199498793887, 4.70230174869687218784713714020, 5.32810490560116582990412818264, 5.72338911596447562569505218039, 6.04798848203660821052489182951, 6.37661064122731610749270783409, 6.97653088727263142756389606857, 7.36963711644089518142039272150
