| L(s) = 1 | + 3-s + 7-s + 9-s − 6·11-s − 6·13-s − 2·19-s + 21-s − 8·23-s + 27-s + 10·29-s + 2·31-s − 6·33-s − 8·37-s − 6·39-s − 6·41-s − 12·43-s − 8·47-s + 49-s − 10·53-s − 2·57-s − 4·59-s + 2·61-s + 63-s − 8·67-s − 8·69-s + 10·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s − 0.458·19-s + 0.218·21-s − 1.66·23-s + 0.192·27-s + 1.85·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s − 0.960·39-s − 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.264·57-s − 0.520·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s − 0.963·69-s + 1.18·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8804664232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8804664232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.04172638807465, −14.35689897602835, −14.03067061853140, −13.52600919378051, −12.84722195458090, −12.43828473767796, −11.91952748847009, −11.38312008029925, −10.37879053164235, −10.11064883017093, −10.00993972299564, −8.978272643489263, −8.311934508894104, −8.004427431361619, −7.594291205941652, −6.813524323605674, −6.297400549937383, −5.333516515218442, −4.802980408586982, −4.611774900804657, −3.421355180156696, −2.938912094814717, −2.177886329392993, −1.799170080384139, −0.3115484902671259,
0.3115484902671259, 1.799170080384139, 2.177886329392993, 2.938912094814717, 3.421355180156696, 4.611774900804657, 4.802980408586982, 5.333516515218442, 6.297400549937383, 6.813524323605674, 7.594291205941652, 8.004427431361619, 8.311934508894104, 8.978272643489263, 10.00993972299564, 10.11064883017093, 10.37879053164235, 11.38312008029925, 11.91952748847009, 12.43828473767796, 12.84722195458090, 13.52600919378051, 14.03067061853140, 14.35689897602835, 15.04172638807465
