| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 4·6-s + 3·9-s + 4·12-s + 8·13-s − 4·16-s − 6·18-s − 16·26-s + 4·27-s + 4·31-s + 8·32-s + 6·36-s + 16·37-s + 16·39-s + 4·41-s + 8·43-s − 8·48-s + 10·49-s + 16·52-s − 12·53-s − 8·54-s − 8·62-s − 8·64-s − 24·67-s + 24·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s + 9-s + 1.15·12-s + 2.21·13-s − 16-s − 1.41·18-s − 3.13·26-s + 0.769·27-s + 0.718·31-s + 1.41·32-s + 36-s + 2.63·37-s + 2.56·39-s + 0.624·41-s + 1.21·43-s − 1.15·48-s + 10/7·49-s + 2.21·52-s − 1.64·53-s − 1.08·54-s − 1.01·62-s − 64-s − 2.93·67-s + 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.690659418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.690659418\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.75558539681191237982733502033, −10.37572245916199996434400696749, −9.731919379183810237879646939798, −9.565007096933867155428190823519, −9.021310181701036461674528670792, −8.751767550179133193101784773870, −8.300007473630957134859413666282, −8.060124129098374446779750896102, −7.46693574337243420597175725925, −7.27223606196994065841182935122, −6.39758780459616600950820717749, −6.16726686486920548320386403878, −5.62975471066998538591265394359, −4.40202095210629136222690685689, −4.37705712725384056723531666911, −3.64955404740536252393515620589, −2.87443015071539085412668990651, −2.43747873830853088467340306986, −1.43896897148633016040909935147, −1.00918287557639663368206734040,
1.00918287557639663368206734040, 1.43896897148633016040909935147, 2.43747873830853088467340306986, 2.87443015071539085412668990651, 3.64955404740536252393515620589, 4.37705712725384056723531666911, 4.40202095210629136222690685689, 5.62975471066998538591265394359, 6.16726686486920548320386403878, 6.39758780459616600950820717749, 7.27223606196994065841182935122, 7.46693574337243420597175725925, 8.060124129098374446779750896102, 8.300007473630957134859413666282, 8.751767550179133193101784773870, 9.021310181701036461674528670792, 9.565007096933867155428190823519, 9.731919379183810237879646939798, 10.37572245916199996434400696749, 10.75558539681191237982733502033
