| L(s) = 1 | + 1.28·7-s − 5.09·11-s − 3.56·13-s + 0.895·17-s − 5.34·19-s − 5.06·23-s + 3·29-s − 6.59·31-s + 7.24·37-s + 7.84·41-s − 10.9·43-s + 2.96·47-s − 5.34·49-s + 4.78·53-s + 5.75·59-s + 4.34·61-s + 8.53·67-s + 5.34·71-s + 9.34·73-s − 6.56·77-s − 0.741·79-s + 9.21·83-s + 9.24·89-s − 4.59·91-s + 3.45·97-s + 16.4·101-s − 15.8·103-s + ⋯ |
| L(s) = 1 | + 0.486·7-s − 1.53·11-s − 0.990·13-s + 0.217·17-s − 1.22·19-s − 1.05·23-s + 0.557·29-s − 1.18·31-s + 1.19·37-s + 1.22·41-s − 1.66·43-s + 0.433·47-s − 0.762·49-s + 0.657·53-s + 0.748·59-s + 0.555·61-s + 1.04·67-s + 0.633·71-s + 1.09·73-s − 0.747·77-s − 0.0834·79-s + 1.01·83-s + 0.980·89-s − 0.482·91-s + 0.351·97-s + 1.63·101-s − 1.56·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.251562688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.251562688\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 0.895T + 17T^{2} \) |
| 19 | \( 1 + 5.34T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 - 7.24T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 - 8.53T + 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 + 0.741T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 9.24T + 89T^{2} \) |
| 97 | \( 1 - 3.45T + 97T^{2} \) |
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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
−8.038198629796821277047847399504, −7.24718118686103054496144209035, −6.45134200702163617680319062651, −5.63093082743812055295637974315, −5.03400444103882579094999982728, −4.41803051352315523123854265901, −3.51136340336295772418457510184, −2.40470131182841816736455134508, −2.08235936036677923486233799152, −0.52062782732269794581929325994,
0.52062782732269794581929325994, 2.08235936036677923486233799152, 2.40470131182841816736455134508, 3.51136340336295772418457510184, 4.41803051352315523123854265901, 5.03400444103882579094999982728, 5.63093082743812055295637974315, 6.45134200702163617680319062651, 7.24718118686103054496144209035, 8.038198629796821277047847399504
