| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 13-s − 2·14-s + 16-s + 2·19-s − 20-s + 25-s + 26-s − 2·28-s − 6·29-s − 5·31-s + 32-s + 2·35-s − 2·37-s + 2·38-s − 40-s − 6·41-s + 8·43-s + 6·47-s − 3·49-s + 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.458·19-s − 0.223·20-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.11·29-s − 0.898·31-s + 0.176·32-s + 0.338·35-s − 0.328·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807694720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807694720\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−12.59342352990830, −12.16189084768502, −11.83098253804459, −11.21624093716884, −10.86197337191845, −10.48208569408266, −9.855320134068374, −9.312578014450769, −9.100118132702629, −8.329232109707636, −7.936919196724093, −7.355558947560419, −6.937270224289679, −6.626246411473150, −5.889509186531541, −5.506028380303901, −5.193172284675502, −4.345294564066555, −3.917986655215483, −3.630281126967878, −2.941919686690655, −2.576345771886470, −1.769018918164935, −1.206207029796835, −0.3103955348248216,
0.3103955348248216, 1.206207029796835, 1.769018918164935, 2.576345771886470, 2.941919686690655, 3.630281126967878, 3.917986655215483, 4.345294564066555, 5.193172284675502, 5.506028380303901, 5.889509186531541, 6.626246411473150, 6.937270224289679, 7.355558947560419, 7.936919196724093, 8.329232109707636, 9.100118132702629, 9.312578014450769, 9.855320134068374, 10.48208569408266, 10.86197337191845, 11.21624093716884, 11.83098253804459, 12.16189084768502, 12.59342352990830
