| L(s) = 1 | − 2-s + 3-s − 6-s + 7-s + 8-s − 10·13-s − 14-s − 16-s − 6·17-s + 7·19-s + 21-s − 6·23-s + 24-s + 10·26-s − 27-s − 8·31-s + 6·34-s − 37-s − 7·38-s − 10·39-s − 42-s − 16·43-s + 6·46-s + 6·47-s − 48-s − 6·49-s − 6·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2.77·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 1.60·19-s + 0.218·21-s − 1.25·23-s + 0.204·24-s + 1.96·26-s − 0.192·27-s − 1.43·31-s + 1.02·34-s − 0.164·37-s − 1.13·38-s − 1.60·39-s − 0.154·42-s − 2.43·43-s + 0.884·46-s + 0.875·47-s − 0.144·48-s − 6/7·49-s − 0.840·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6355574783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6355574783\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−10.02703468974041877185251715036, −9.525766818303533896474423004274, −9.239803040726290765750313419768, −9.204933924759514965685025924644, −8.405253032360885644011994647749, −7.88753918827989874547607118196, −7.87149881954552970509503110321, −7.38364876412076295594692138021, −6.81607552527801770307819077713, −6.67215301209161030783520210664, −5.82521485263371149903106564716, −5.18095264175520696050491607986, −4.79711603892737432456889712207, −4.77488490657734986152979262426, −3.59435046012128353016069701723, −3.57009651550726298748689200091, −2.45169380120769233031150109519, −2.26253071491922606438835554724, −1.67875849839359839785987381729, −0.37162367235802753064765556360,
0.37162367235802753064765556360, 1.67875849839359839785987381729, 2.26253071491922606438835554724, 2.45169380120769233031150109519, 3.57009651550726298748689200091, 3.59435046012128353016069701723, 4.77488490657734986152979262426, 4.79711603892737432456889712207, 5.18095264175520696050491607986, 5.82521485263371149903106564716, 6.67215301209161030783520210664, 6.81607552527801770307819077713, 7.38364876412076295594692138021, 7.87149881954552970509503110321, 7.88753918827989874547607118196, 8.405253032360885644011994647749, 9.204933924759514965685025924644, 9.239803040726290765750313419768, 9.525766818303533896474423004274, 10.02703468974041877185251715036
