| L(s) = 1 | + 5-s − 4.21·7-s − 5.57·11-s − 2.21·13-s − 0.643·17-s − 19-s − 1.35·23-s + 25-s − 4.86·29-s + 2.64·31-s − 4.21·35-s − 2.21·37-s − 3.57·41-s − 0.218·43-s + 1.35·47-s + 10.7·49-s − 0.643·53-s − 5.57·55-s + 4.43·59-s + 13.0·61-s − 2.21·65-s + 1.28·67-s − 11.1·71-s + 10·73-s + 23.5·77-s − 1.28·79-s − 12.3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.59·7-s − 1.68·11-s − 0.615·13-s − 0.156·17-s − 0.229·19-s − 0.282·23-s + 0.200·25-s − 0.902·29-s + 0.474·31-s − 0.713·35-s − 0.364·37-s − 0.558·41-s − 0.0332·43-s + 0.197·47-s + 1.54·49-s − 0.0883·53-s − 0.751·55-s + 0.577·59-s + 1.67·61-s − 0.275·65-s + 0.157·67-s − 1.32·71-s + 1.17·73-s + 2.67·77-s − 0.144·79-s − 1.35·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7604653978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7604653978\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 + 0.643T + 17T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 + 4.86T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 + 2.21T + 37T^{2} \) |
| 41 | \( 1 + 3.57T + 41T^{2} \) |
| 43 | \( 1 + 0.218T + 43T^{2} \) |
| 47 | \( 1 - 1.35T + 47T^{2} \) |
| 53 | \( 1 + 0.643T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 1.28T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 5.78T + 97T^{2} \) |
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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
−7.935720455759135714601586809692, −7.14172961724583404182000222020, −6.64707547126954858779869498047, −5.74646755416260730457685058246, −5.35749314598316512908661162913, −4.38367525189307616810245597404, −3.39635178657277739255259027025, −2.73639661592754042188649562879, −2.06408625551079649802594765511, −0.41199573048914462202181068889,
0.41199573048914462202181068889, 2.06408625551079649802594765511, 2.73639661592754042188649562879, 3.39635178657277739255259027025, 4.38367525189307616810245597404, 5.35749314598316512908661162913, 5.74646755416260730457685058246, 6.64707547126954858779869498047, 7.14172961724583404182000222020, 7.935720455759135714601586809692
