| L(s) = 1 | + 5-s + 3.37·7-s + 3.37·11-s − 13-s + 1.37·17-s − 6.74·19-s + 0.627·23-s + 25-s + 2·29-s + 6.74·31-s + 3.37·35-s + 5.37·37-s + 1.37·41-s − 4·43-s + 1.25·47-s + 4.37·49-s + 9.37·53-s + 3.37·55-s + 8·59-s + 8.11·61-s − 65-s − 4·67-s + 11.3·71-s − 15.4·73-s + 11.3·77-s − 16.8·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.27·7-s + 1.01·11-s − 0.277·13-s + 0.332·17-s − 1.54·19-s + 0.130·23-s + 0.200·25-s + 0.371·29-s + 1.21·31-s + 0.570·35-s + 0.883·37-s + 0.214·41-s − 0.609·43-s + 0.183·47-s + 0.624·49-s + 1.28·53-s + 0.454·55-s + 1.04·59-s + 1.03·61-s − 0.124·65-s − 0.488·67-s + 1.34·71-s − 1.81·73-s + 1.29·77-s − 1.89·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.743217492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743217492\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 0.627T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 9.37T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 2.62T + 97T^{2} \) |
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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
−8.539591034799263776642817151218, −7.62361445424448438002783506348, −6.79457541171731475200188367939, −6.17109484836271941601991750976, −5.33570198039926317985809336004, −4.51957651184034914620654316527, −4.01449283600559416858645553069, −2.69790451930545950783176095771, −1.88397161395202706320366755395, −0.981878954108357832189789184523,
0.981878954108357832189789184523, 1.88397161395202706320366755395, 2.69790451930545950783176095771, 4.01449283600559416858645553069, 4.51957651184034914620654316527, 5.33570198039926317985809336004, 6.17109484836271941601991750976, 6.79457541171731475200188367939, 7.62361445424448438002783506348, 8.539591034799263776642817151218
