| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s + 17-s + 18-s − 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s + 17-s + 18-s − 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.999656238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.999656238\) |
| \(L(1)\) |
\(\approx\) |
\(3.048191033\) |
| \(L(1)\) |
\(\approx\) |
\(3.048191033\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 239 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.60508485977741275278592510674, −25.1163674104731750738621505485, −24.36676486814668059717776604754, −23.10790676955752710928937839444, −21.99084946154301647258002202260, −21.58598211259357641224973940182, −20.51141316993130405142799340128, −19.61856079967804902450615535754, −18.98032740736430143908758825323, −17.27397609663823143706270924439, −16.40688356688606174307839529215, −15.2608509097296805646480490650, −14.32021953735057321817384874797, −13.81961781520180592123975452812, −12.756134637546639276007825493374, −12.0960240955773604076310168012, −10.16170750097920561235259108956, −9.79002055879142391592879804796, −8.36294248505074812308056341262, −6.894552102421987626837985139766, −6.28165603793688808739543953718, −4.8417909650967718195071232333, −3.60622840196448289993570005114, −2.669124478877619292466489390, −1.60129099812799297030028865969,
1.60129099812799297030028865969, 2.669124478877619292466489390, 3.60622840196448289993570005114, 4.8417909650967718195071232333, 6.28165603793688808739543953718, 6.894552102421987626837985139766, 8.36294248505074812308056341262, 9.79002055879142391592879804796, 10.16170750097920561235259108956, 12.0960240955773604076310168012, 12.756134637546639276007825493374, 13.81961781520180592123975452812, 14.32021953735057321817384874797, 15.2608509097296805646480490650, 16.40688356688606174307839529215, 17.27397609663823143706270924439, 18.98032740736430143908758825323, 19.61856079967804902450615535754, 20.51141316993130405142799340128, 21.58598211259357641224973940182, 21.99084946154301647258002202260, 23.10790676955752710928937839444, 24.36676486814668059717776604754, 25.1163674104731750738621505485, 25.60508485977741275278592510674
