| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 11-s − 13-s + 15-s − 17-s + 19-s + 21-s − 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 35-s + 37-s − 39-s − 43-s + 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + 57-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 11-s − 13-s + 15-s − 17-s + 19-s + 21-s − 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 35-s + 37-s − 39-s − 43-s + 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.495248003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.495248003\) |
| \(L(1)\) |
\(\approx\) |
\(1.962537372\) |
| \(L(1)\) |
\(\approx\) |
\(1.962537372\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 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 | \( 1 + T \) |
| 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
−27.18959889827525835507606133049, −26.556573661788412799274004730151, −25.41787487293047390063255639176, −24.596231297501828865578001375619, −24.14327129916267094557022012853, −22.08985496565636137713134365135, −21.76871140328820111848253409537, −20.40075579065797120810936460854, −19.98766130452250258963095933268, −18.518338453375829703919243506631, −17.7226514273478932989914812752, −16.69268647565250647143252188365, −15.168058966676505331879804974100, −14.34262395814829102597530018745, −13.74885643804074813326615353587, −12.50600873575682915211648357070, −11.17523921276331737183421168744, −9.75749111863719037075317777730, −9.13366803346812434944123442529, −7.893913960228264342361006638520, −6.78498054857503226782447673144, −5.247872895935766107634003439614, −4.02686663076884335176349926948, −2.400721649212977370405088985897, −1.52681653355406761186483111055,
1.52681653355406761186483111055, 2.400721649212977370405088985897, 4.02686663076884335176349926948, 5.247872895935766107634003439614, 6.78498054857503226782447673144, 7.893913960228264342361006638520, 9.13366803346812434944123442529, 9.75749111863719037075317777730, 11.17523921276331737183421168744, 12.50600873575682915211648357070, 13.74885643804074813326615353587, 14.34262395814829102597530018745, 15.168058966676505331879804974100, 16.69268647565250647143252188365, 17.7226514273478932989914812752, 18.518338453375829703919243506631, 19.98766130452250258963095933268, 20.40075579065797120810936460854, 21.76871140328820111848253409537, 22.08985496565636137713134365135, 24.14327129916267094557022012853, 24.596231297501828865578001375619, 25.41787487293047390063255639176, 26.556573661788412799274004730151, 27.18959889827525835507606133049
