| L(s) = 1 | + 3-s + 9-s + 11-s − 13-s + 17-s − 19-s + 23-s + 27-s − 29-s + 31-s + 33-s + 37-s − 39-s − 41-s − 43-s − 47-s + 51-s + 53-s − 57-s − 59-s + 61-s − 67-s + 69-s − 71-s + 73-s − 79-s + 81-s + ⋯ |
| L(s) = 1 | + 3-s + 9-s + 11-s − 13-s + 17-s − 19-s + 23-s + 27-s − 29-s + 31-s + 33-s + 37-s − 39-s − 41-s − 43-s − 47-s + 51-s + 53-s − 57-s − 59-s + 61-s − 67-s + 69-s − 71-s + 73-s − 79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.800883113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800883113\) |
| \(L(1)\) |
\(\approx\) |
\(1.486528806\) |
| \(L(1)\) |
\(\approx\) |
\(1.486528806\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 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
−25.43516320010697102271833828648, −24.93996550181417715382418421401, −24.01594057573158263298849860487, −22.8987792524134657539146383856, −21.795266510692869098169201177216, −21.085251664604803758793852989412, −20.04378853339942956201837048036, −19.32210955615113433986353993220, −18.62360318272331579959194053137, −17.21961794215015108163019293942, −16.50929970985619618057322434766, −14.90752734810907273966102792006, −14.82181566500290541463541840892, −13.59023798348914133920482719044, −12.65197059597322821282512913920, −11.660991001209588227444650229069, −10.22598963219128912389025130927, −9.436556029811699634783467907818, −8.48376317055071607937249620412, −7.45909259806219887737663429772, −6.49885154025400085935019767082, −4.91342919831697486359069873108, −3.81693581579519539571082478071, −2.72772497869955935520579426490, −1.46809557272934852731422834119,
1.46809557272934852731422834119, 2.72772497869955935520579426490, 3.81693581579519539571082478071, 4.91342919831697486359069873108, 6.49885154025400085935019767082, 7.45909259806219887737663429772, 8.48376317055071607937249620412, 9.436556029811699634783467907818, 10.22598963219128912389025130927, 11.660991001209588227444650229069, 12.65197059597322821282512913920, 13.59023798348914133920482719044, 14.82181566500290541463541840892, 14.90752734810907273966102792006, 16.50929970985619618057322434766, 17.21961794215015108163019293942, 18.62360318272331579959194053137, 19.32210955615113433986353993220, 20.04378853339942956201837048036, 21.085251664604803758793852989412, 21.795266510692869098169201177216, 22.8987792524134657539146383856, 24.01594057573158263298849860487, 24.93996550181417715382418421401, 25.43516320010697102271833828648
