| L(s) = 1 | + 2-s − 4-s − 2·5-s − 2·7-s − 3·8-s − 2·10-s + 11-s − 13-s − 2·14-s − 16-s + 4·17-s + 6·19-s + 2·20-s + 22-s + 8·23-s − 25-s − 26-s + 2·28-s + 10·31-s + 5·32-s + 4·34-s + 4·35-s − 6·37-s + 6·38-s + 6·40-s − 2·41-s − 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.755·7-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 0.277·13-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 1.37·19-s + 0.447·20-s + 0.213·22-s + 1.66·23-s − 1/5·25-s − 0.196·26-s + 0.377·28-s + 1.79·31-s + 0.883·32-s + 0.685·34-s + 0.676·35-s − 0.986·37-s + 0.973·38-s + 0.948·40-s − 0.312·41-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.429982381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.429982381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.573421641962975893437284232846, −8.957347110213973592791803261844, −7.966905114856647208944346582611, −7.18584010360087641019743865713, −6.23113991377994761358953804072, −5.27061465903248153356902936682, −4.53948320527620847342463349489, −3.44138640243946987623938330137, −3.06772105527612509209487610972, −0.799477074197494500154640207567,
0.799477074197494500154640207567, 3.06772105527612509209487610972, 3.44138640243946987623938330137, 4.53948320527620847342463349489, 5.27061465903248153356902936682, 6.23113991377994761358953804072, 7.18584010360087641019743865713, 7.966905114856647208944346582611, 8.957347110213973592791803261844, 9.573421641962975893437284232846
