L(s) = 1 | − 2·3-s − 5-s + 3·9-s + 11-s + 6·13-s + 2·15-s − 2·17-s − 5·19-s − 7·23-s − 10·27-s − 12·29-s + 4·31-s − 2·33-s + 5·37-s − 12·39-s + 10·41-s + 12·43-s − 3·45-s − 9·47-s + 4·51-s − 11·53-s − 55-s + 10·57-s + 8·59-s − 12·61-s − 6·65-s + 4·67-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.485·17-s − 1.14·19-s − 1.45·23-s − 1.92·27-s − 2.22·29-s + 0.718·31-s − 0.348·33-s + 0.821·37-s − 1.92·39-s + 1.56·41-s + 1.82·43-s − 0.447·45-s − 1.31·47-s + 0.560·51-s − 1.51·53-s − 0.134·55-s + 1.32·57-s + 1.04·59-s − 1.53·61-s − 0.744·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8219902123\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8219902123\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.321651539676489981663414754950, −9.134081321630978594977132083773, −8.428784275500972841118872782813, −8.306200387651408987341462356947, −7.63102494361054254692186417646, −7.53689981091878564986593317378, −7.01725939711515055639824937836, −6.38085062405541775631951690032, −6.04368449582268823240536042515, −5.90861127441862910397381835284, −5.73002376479802169829029793518, −4.77631265998901889030915020096, −4.33118417454365400517022232495, −4.28081560281998341919468480497, −3.50311324456330301325603241294, −3.45768349910297191515617908531, −2.13006107416386283590282070532, −2.04064952679526463951370381090, −1.17397570701143918203395662333, −0.38991862446237965611773256504,
0.38991862446237965611773256504, 1.17397570701143918203395662333, 2.04064952679526463951370381090, 2.13006107416386283590282070532, 3.45768349910297191515617908531, 3.50311324456330301325603241294, 4.28081560281998341919468480497, 4.33118417454365400517022232495, 4.77631265998901889030915020096, 5.73002376479802169829029793518, 5.90861127441862910397381835284, 6.04368449582268823240536042515, 6.38085062405541775631951690032, 7.01725939711515055639824937836, 7.53689981091878564986593317378, 7.63102494361054254692186417646, 8.306200387651408987341462356947, 8.428784275500972841118872782813, 9.134081321630978594977132083773, 9.321651539676489981663414754950