| L(s) = 1 | + 7-s + 3·11-s − 2·13-s + 3·17-s + 7·19-s + 6·29-s + 4·31-s − 8·37-s + 9·41-s + 8·43-s + 6·47-s + 49-s − 12·53-s + 12·59-s − 10·61-s − 7·67-s + 6·71-s − 5·73-s + 3·77-s − 14·79-s + 9·83-s + 15·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.904·11-s − 0.554·13-s + 0.727·17-s + 1.60·19-s + 1.11·29-s + 0.718·31-s − 1.31·37-s + 1.40·41-s + 1.21·43-s + 0.875·47-s + 1/7·49-s − 1.64·53-s + 1.56·59-s − 1.28·61-s − 0.855·67-s + 0.712·71-s − 0.585·73-s + 0.341·77-s − 1.57·79-s + 0.987·83-s + 1.58·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.102491139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.102491139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.49317589080920, −14.52144987133278, −14.36657061909992, −13.93746542658737, −13.31983108142559, −12.39836370254997, −12.17377199595095, −11.68373156901627, −11.10026102623609, −10.36296383875590, −9.935071939859098, −9.258984740241687, −8.886626180090970, −8.080269680030353, −7.472171344218041, −7.168120694180656, −6.233228967849959, −5.810816175941962, −4.968992336192474, −4.571645297465133, −3.695667476592717, −3.097365787500898, −2.340437744706583, −1.339126179876382, −0.7857888990684894,
0.7857888990684894, 1.339126179876382, 2.340437744706583, 3.097365787500898, 3.695667476592717, 4.571645297465133, 4.968992336192474, 5.810816175941962, 6.233228967849959, 7.168120694180656, 7.472171344218041, 8.080269680030353, 8.886626180090970, 9.258984740241687, 9.935071939859098, 10.36296383875590, 11.10026102623609, 11.68373156901627, 12.17377199595095, 12.39836370254997, 13.31983108142559, 13.93746542658737, 14.36657061909992, 14.52144987133278, 15.49317589080920
