| L(s) = 1 | + 2-s + 4-s + 3·5-s − 4·7-s + 8-s + 3·10-s − 13-s − 4·14-s + 16-s + 3·17-s − 4·19-s + 3·20-s + 4·25-s − 26-s − 4·28-s − 9·29-s − 4·31-s + 32-s + 3·34-s − 12·35-s − 37-s − 4·38-s + 3·40-s − 6·41-s + 8·43-s + 12·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.51·7-s + 0.353·8-s + 0.948·10-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.727·17-s − 0.917·19-s + 0.670·20-s + 4/5·25-s − 0.196·26-s − 0.755·28-s − 1.67·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 2.02·35-s − 0.164·37-s − 0.648·38-s + 0.474·40-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736780166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736780166\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−12.97130443985463931376959920846, −12.30257690637103727384354789287, −10.74648774185021180091471784521, −9.880068252452829064500103261921, −9.101601246278505892226541271616, −7.24345462277249174674167705434, −6.18735538198950166736525445007, −5.49090131952205180218546644094, −3.69589098584017718744945020548, −2.30016225881827778465495235815,
2.30016225881827778465495235815, 3.69589098584017718744945020548, 5.49090131952205180218546644094, 6.18735538198950166736525445007, 7.24345462277249174674167705434, 9.101601246278505892226541271616, 9.880068252452829064500103261921, 10.74648774185021180091471784521, 12.30257690637103727384354789287, 12.97130443985463931376959920846
