| L(s) = 1 | + 2·3-s − 4·4-s + 3·9-s − 8·12-s + 12·16-s − 12·23-s + 4·27-s + 2·31-s − 12·36-s + 10·37-s + 24·47-s + 24·48-s − 11·49-s − 12·53-s − 32·64-s + 10·67-s − 24·69-s − 12·71-s + 5·81-s − 12·89-s + 48·92-s + 4·93-s + 26·97-s + 26·103-s − 16·108-s + 20·111-s − 8·124-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s + 9-s − 2.30·12-s + 3·16-s − 2.50·23-s + 0.769·27-s + 0.359·31-s − 2·36-s + 1.64·37-s + 3.50·47-s + 3.46·48-s − 1.57·49-s − 1.64·53-s − 4·64-s + 1.22·67-s − 2.88·69-s − 1.42·71-s + 5/9·81-s − 1.27·89-s + 5.00·92-s + 0.414·93-s + 2.63·97-s + 2.56·103-s − 1.53·108-s + 1.89·111-s − 0.718·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.904343065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.904343065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−8.032054782214698978927112473398, −7.70675399911308994916913563422, −7.43563381746346935311325609793, −7.13711717080999035317305935993, −6.26749395255807075850422212455, −6.22958245868092253509088806722, −5.80486623397408439772546422653, −5.54950452968540898834852899590, −4.88504411183406661485358194193, −4.69339965261509922777148182097, −4.27044203224783758105407073860, −4.12936962423859347972747161874, −3.67925688564252219474166957945, −3.45397851510919041401784544714, −2.81981699177738900059336227282, −2.61428926486940176230141930942, −1.74944808436051574548016877352, −1.72958462479312913654454533508, −0.68543110405201829306145238655, −0.56602574337960999418448541241,
0.56602574337960999418448541241, 0.68543110405201829306145238655, 1.72958462479312913654454533508, 1.74944808436051574548016877352, 2.61428926486940176230141930942, 2.81981699177738900059336227282, 3.45397851510919041401784544714, 3.67925688564252219474166957945, 4.12936962423859347972747161874, 4.27044203224783758105407073860, 4.69339965261509922777148182097, 4.88504411183406661485358194193, 5.54950452968540898834852899590, 5.80486623397408439772546422653, 6.22958245868092253509088806722, 6.26749395255807075850422212455, 7.13711717080999035317305935993, 7.43563381746346935311325609793, 7.70675399911308994916913563422, 8.032054782214698978927112473398
