| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 11-s + 2·13-s + 16-s + 3·17-s + 2·19-s − 3·20-s − 22-s − 3·23-s + 4·25-s − 2·26-s + 2·31-s − 32-s − 3·34-s + 8·37-s − 2·38-s + 3·40-s + 9·41-s − 4·43-s + 44-s + 3·46-s − 3·47-s − 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 0.670·20-s − 0.213·22-s − 0.625·23-s + 4/5·25-s − 0.392·26-s + 0.359·31-s − 0.176·32-s − 0.514·34-s + 1.31·37-s − 0.324·38-s + 0.474·40-s + 1.40·41-s − 0.609·43-s + 0.150·44-s + 0.442·46-s − 0.437·47-s − 0.565·50-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
| \(L(1)\) |
\(\approx\) |
\(1.035068702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035068702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70568396368074032136470678510, −7.32475867031080814209612554549, −6.39371790783118345004403033853, −5.85107407193854462317294306387, −4.81908334791639931418960930332, −4.03423157696245935221029134224, −3.45465563523040557671191999558, −2.64600941070577403570016606989, −1.42905787877392315938432351420, −0.58025390721148013899741029319,
0.58025390721148013899741029319, 1.42905787877392315938432351420, 2.64600941070577403570016606989, 3.45465563523040557671191999558, 4.03423157696245935221029134224, 4.81908334791639931418960930332, 5.85107407193854462317294306387, 6.39371790783118345004403033853, 7.32475867031080814209612554549, 7.70568396368074032136470678510
