| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s − 3·9-s + 6·11-s + 13-s − 4·14-s − 16-s + 6·17-s − 3·18-s + 4·19-s + 6·22-s − 5·25-s + 26-s + 4·28-s + 6·29-s + 6·31-s + 5·32-s + 6·34-s + 3·36-s + 4·37-s + 4·38-s − 10·41-s + 8·43-s − 6·44-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s − 9-s + 1.80·11-s + 0.277·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.917·19-s + 1.27·22-s − 25-s + 0.196·26-s + 0.755·28-s + 1.11·29-s + 1.07·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s + 0.657·37-s + 0.648·38-s − 1.56·41-s + 1.21·43-s − 0.904·44-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.645687224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.645687224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−9.574522541642589589985188016794, −8.988998659278615284491246060023, −8.126070510339494879263325441746, −6.83925221987190563365247183345, −6.06474820112384167171134378685, −5.65623625948665695335686080126, −4.33137479137442781148439340290, −3.45764830979661882617540127189, −2.99858257764041482356819499984, −0.856282992686243285446003705550,
0.856282992686243285446003705550, 2.99858257764041482356819499984, 3.45764830979661882617540127189, 4.33137479137442781148439340290, 5.65623625948665695335686080126, 6.06474820112384167171134378685, 6.83925221987190563365247183345, 8.126070510339494879263325441746, 8.988998659278615284491246060023, 9.574522541642589589985188016794
