L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 11-s − 2·13-s − 2·14-s + 16-s − 6·17-s + 22-s − 5·25-s + 2·26-s + 2·28-s + 6·29-s + 4·31-s − 32-s + 6·34-s − 2·37-s − 6·41-s − 10·43-s − 44-s + 12·47-s − 3·49-s + 5·50-s − 2·52-s + 12·53-s − 2·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 0.301·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.213·22-s − 25-s + 0.392·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.937·41-s − 1.52·43-s − 0.150·44-s + 1.75·47-s − 3/7·49-s + 0.707·50-s − 0.277·52-s + 1.64·53-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71478 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71478 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7743457725\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7743457725\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.02690139648464, −13.53277223324984, −13.38619095106936, −12.28546219474758, −12.10607307095972, −11.60133892132680, −11.01648610754806, −10.52364396674828, −10.16430442608447, −9.544912052728102, −8.934485982896296, −8.532314175213672, −8.022316376307488, −7.568254500073955, −6.864317257394312, −6.559888663863113, −5.770918705118921, −5.213581175370027, −4.503027401265414, −4.199006888556705, −3.128264905378774, −2.600295318368729, −1.921310645516587, −1.386615466535308, −0.3212033568098682,
0.3212033568098682, 1.386615466535308, 1.921310645516587, 2.600295318368729, 3.128264905378774, 4.199006888556705, 4.503027401265414, 5.213581175370027, 5.770918705118921, 6.559888663863113, 6.864317257394312, 7.568254500073955, 8.022316376307488, 8.532314175213672, 8.934485982896296, 9.544912052728102, 10.16430442608447, 10.52364396674828, 11.01648610754806, 11.60133892132680, 12.10607307095972, 12.28546219474758, 13.38619095106936, 13.53277223324984, 14.02690139648464