L(s) = 1 | + 2-s − 4-s − 4·5-s + 2·7-s − 3·8-s − 4·10-s − 11-s − 2·13-s + 2·14-s − 16-s + 2·17-s + 6·19-s + 4·20-s − 22-s + 11·25-s − 2·26-s − 2·28-s + 6·29-s + 4·31-s + 5·32-s + 2·34-s − 8·35-s + 6·37-s + 6·38-s + 12·40-s + 10·41-s − 6·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s + 0.755·7-s − 1.06·8-s − 1.26·10-s − 0.301·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s + 1.37·19-s + 0.894·20-s − 0.213·22-s + 11/5·25-s − 0.392·26-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s − 1.35·35-s + 0.986·37-s + 0.973·38-s + 1.89·40-s + 1.56·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.166200353\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.166200353\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.44443900081476, −14.07452785124187, −13.52036604636232, −12.84560668971391, −12.32637545566918, −11.94164837823176, −11.66062130884486, −11.09428316069912, −10.46026465847278, −9.784715747940286, −9.231978948385460, −8.554654239814450, −8.083503775601112, −7.675406842973844, −7.286698487925250, −6.447361005628669, −5.721617602333966, −5.066691782691363, −4.597409487051051, −4.334993964818023, −3.493221681689897, −3.121289701664148, −2.434202604643444, −1.027226569616597, −0.5810932886573262,
0.5810932886573262, 1.027226569616597, 2.434202604643444, 3.121289701664148, 3.493221681689897, 4.334993964818023, 4.597409487051051, 5.066691782691363, 5.721617602333966, 6.447361005628669, 7.286698487925250, 7.675406842973844, 8.083503775601112, 8.554654239814450, 9.231978948385460, 9.784715747940286, 10.46026465847278, 11.09428316069912, 11.66062130884486, 11.94164837823176, 12.32637545566918, 12.84560668971391, 13.52036604636232, 14.07452785124187, 14.44443900081476