| L(s) = 1 | − 2.48·2-s + 4.15·4-s + 2.80·7-s − 5.35·8-s + 0.675·11-s + 6.63·13-s − 6.96·14-s + 4.96·16-s + 2.83·17-s + 3·19-s − 1.67·22-s − 1.32·23-s − 16.4·26-s + 11.6·28-s − 0.906·29-s + 31-s − 1.61·32-s − 7.02·34-s + 0.712·37-s − 7.44·38-s + 4.21·41-s + 0.775·43-s + 2.80·44-s + 3.28·46-s + 3.03·47-s + 0.873·49-s + 27.5·52-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 2.07·4-s + 1.06·7-s − 1.89·8-s + 0.203·11-s + 1.84·13-s − 1.86·14-s + 1.24·16-s + 0.686·17-s + 0.688·19-s − 0.357·22-s − 0.276·23-s − 3.22·26-s + 2.20·28-s − 0.168·29-s + 0.179·31-s − 0.284·32-s − 1.20·34-s + 0.117·37-s − 1.20·38-s + 0.658·41-s + 0.118·43-s + 0.423·44-s + 0.484·46-s + 0.443·47-s + 0.124·49-s + 3.82·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.303951546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.303951546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 - 0.675T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 + 0.906T + 29T^{2} \) |
| 37 | \( 1 - 0.712T + 37T^{2} \) |
| 41 | \( 1 - 4.21T + 41T^{2} \) |
| 43 | \( 1 - 0.775T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 + 1.86T + 79T^{2} \) |
| 83 | \( 1 - 7.95T + 83T^{2} \) |
| 89 | \( 1 - 6.67T + 89T^{2} \) |
| 97 | \( 1 + 4.50T + 97T^{2} \) |
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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
−8.053216352951155173599644335042, −7.60348690792695699046130886392, −6.82682696850298276513664953130, −6.04199944283146971207984025718, −5.41169259019449137488827309704, −4.24031518038589890799523540922, −3.37926872636505339908936220495, −2.28735092345474814453979899205, −1.38309922256865639846575437420, −0.885308570468560949032924052228,
0.885308570468560949032924052228, 1.38309922256865639846575437420, 2.28735092345474814453979899205, 3.37926872636505339908936220495, 4.24031518038589890799523540922, 5.41169259019449137488827309704, 6.04199944283146971207984025718, 6.82682696850298276513664953130, 7.60348690792695699046130886392, 8.053216352951155173599644335042
