| 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 + 33-s − 35-s − 37-s − 39-s + 41-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 + 33-s − 35-s − 37-s − 39-s + 41-s + 43-s + 45-s − 47-s + 49-s − 51-s − 53-s + 55-s − 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.521673262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.521673262\) |
| \(L(1)\) |
\(\approx\) |
\(1.440364958\) |
| \(L(1)\) |
\(\approx\) |
\(1.440364958\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 \) |
| 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 | \( 1 - T \) |
| 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
−29.334871091113484946351137707572, −27.86989177405450835475666477964, −26.643425970432126302082709821967, −25.880371750170322964315511561254, −25.00630434312822634580062344622, −24.356752363254936795830263546196, −22.56680147783657639155465216822, −21.837061891821094086868452198781, −20.78057473130457487344629636083, −19.63238815203362026200531924534, −19.0476706859576487353641709858, −17.57404979005130372970433458233, −16.625330402686076544854625120455, −15.213111424522225983101782724, −14.34174965539024265120878009326, −13.27145934970391320424943428915, −12.539618936619980847494583332012, −10.61990372796675657964321451580, −9.40675792843507810426747657358, −9.00662360321370473931774529159, −7.181539063758576379240514115536, −6.26727016945472537926517796022, −4.48229287568604351517698650740, −3.03206903567326412510611263824, −1.888315542865292829163666093299,
1.888315542865292829163666093299, 3.03206903567326412510611263824, 4.48229287568604351517698650740, 6.26727016945472537926517796022, 7.181539063758576379240514115536, 9.00662360321370473931774529159, 9.40675792843507810426747657358, 10.61990372796675657964321451580, 12.539618936619980847494583332012, 13.27145934970391320424943428915, 14.34174965539024265120878009326, 15.213111424522225983101782724, 16.625330402686076544854625120455, 17.57404979005130372970433458233, 19.0476706859576487353641709858, 19.63238815203362026200531924534, 20.78057473130457487344629636083, 21.837061891821094086868452198781, 22.56680147783657639155465216822, 24.356752363254936795830263546196, 25.00630434312822634580062344622, 25.880371750170322964315511561254, 26.643425970432126302082709821967, 27.86989177405450835475666477964, 29.334871091113484946351137707572
