| L(s) = 1 | − 0.427·3-s + 4.24·5-s + 7-s − 2.81·9-s + 5.28·11-s − 1.75·13-s − 1.81·15-s − 5.39·17-s + 1.70·19-s − 0.427·21-s − 7.21·23-s + 13.0·25-s + 2.48·27-s − 2.73·29-s + 6.82·31-s − 2.25·33-s + 4.24·35-s + 9.27·37-s + 0.752·39-s + 5.63·41-s + 9.15·43-s − 11.9·45-s + 4.80·47-s + 49-s + 2.30·51-s − 6.25·53-s + 22.4·55-s + ⋯ |
| L(s) = 1 | − 0.246·3-s + 1.89·5-s + 0.377·7-s − 0.939·9-s + 1.59·11-s − 0.487·13-s − 0.469·15-s − 1.30·17-s + 0.391·19-s − 0.0933·21-s − 1.50·23-s + 2.60·25-s + 0.478·27-s − 0.508·29-s + 1.22·31-s − 0.393·33-s + 0.718·35-s + 1.52·37-s + 0.120·39-s + 0.879·41-s + 1.39·43-s − 1.78·45-s + 0.700·47-s + 0.142·49-s + 0.323·51-s − 0.859·53-s + 3.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.646457805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.646457805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 0.427T + 3T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 + 1.75T + 13T^{2} \) |
| 17 | \( 1 + 5.39T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 + 7.21T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 9.27T + 37T^{2} \) |
| 41 | \( 1 - 5.63T + 41T^{2} \) |
| 43 | \( 1 - 9.15T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 + 6.25T + 53T^{2} \) |
| 59 | \( 1 + 3.77T + 59T^{2} \) |
| 61 | \( 1 + 0.481T + 61T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 - 9.06T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 5.71T + 89T^{2} \) |
| 97 | \( 1 + 7.42T + 97T^{2} \) |
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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
−8.781911996138648556235060317491, −7.891733402966643911612800771971, −6.72852057161590061872753827161, −6.18359639984441321215958559569, −5.81102105957266569842418576890, −4.85051436374720330721554460997, −4.06661923147909209239400200218, −2.63090510499110299725800164163, −2.10102811554397282892380487919, −1.01113275103533354764235852003,
1.01113275103533354764235852003, 2.10102811554397282892380487919, 2.63090510499110299725800164163, 4.06661923147909209239400200218, 4.85051436374720330721554460997, 5.81102105957266569842418576890, 6.18359639984441321215958559569, 6.72852057161590061872753827161, 7.891733402966643911612800771971, 8.781911996138648556235060317491
