L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 11-s + 4·13-s + 16-s − 4·19-s + 2·20-s − 22-s − 4·23-s − 25-s + 4·26-s − 2·29-s + 10·31-s + 32-s − 6·37-s − 4·38-s + 2·40-s − 4·43-s − 44-s − 4·46-s + 10·47-s − 50-s + 4·52-s + 14·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s + 1.10·13-s + 1/4·16-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s + 0.784·26-s − 0.371·29-s + 1.79·31-s + 0.176·32-s − 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.609·43-s − 0.150·44-s − 0.589·46-s + 1.45·47-s − 0.141·50-s + 0.554·52-s + 1.92·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.259558353\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.259558353\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.60662678357074446672630692097, −6.66165493167683007748095504732, −6.28710805779090032434203569574, −5.62199336160611525863427043947, −5.06823717564074315586051720075, −4.06867461412619442474056297162, −3.63174413599045265668328472044, −2.45096584239443641859321352226, −2.03357682397592720278356318451, −0.890566128143998534104277981591,
0.890566128143998534104277981591, 2.03357682397592720278356318451, 2.45096584239443641859321352226, 3.63174413599045265668328472044, 4.06867461412619442474056297162, 5.06823717564074315586051720075, 5.62199336160611525863427043947, 6.28710805779090032434203569574, 6.66165493167683007748095504732, 7.60662678357074446672630692097