| L(s) = 1 | + 0.710·2-s + 2.40·3-s − 1.49·4-s + 1.28·5-s + 1.71·6-s + 7-s − 2.48·8-s + 2.79·9-s + 0.916·10-s + 2.40·11-s − 3.60·12-s + 0.710·14-s + 3.10·15-s + 1.22·16-s + 3.90·17-s + 1.98·18-s + 5.89·19-s − 1.92·20-s + 2.40·21-s + 1.71·22-s − 6.32·23-s − 5.97·24-s − 3.33·25-s − 0.486·27-s − 1.49·28-s + 5.61·29-s + 2.20·30-s + ⋯ |
| L(s) = 1 | + 0.502·2-s + 1.39·3-s − 0.747·4-s + 0.576·5-s + 0.698·6-s + 0.377·7-s − 0.877·8-s + 0.932·9-s + 0.289·10-s + 0.726·11-s − 1.03·12-s + 0.189·14-s + 0.801·15-s + 0.306·16-s + 0.946·17-s + 0.468·18-s + 1.35·19-s − 0.431·20-s + 0.525·21-s + 0.364·22-s − 1.31·23-s − 1.22·24-s − 0.667·25-s − 0.0936·27-s − 0.282·28-s + 1.04·29-s + 0.402·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.281484470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.281484470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.710T + 2T^{2} \) |
| 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 17 | \( 1 - 3.90T + 17T^{2} \) |
| 19 | \( 1 - 5.89T + 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 - 7.78T + 41T^{2} \) |
| 43 | \( 1 - 0.289T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 4.03T + 59T^{2} \) |
| 61 | \( 1 + 4.60T + 61T^{2} \) |
| 67 | \( 1 + 7.57T + 67T^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 - 0.899T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−9.529025836474126299504449375420, −9.120226310956534660879786635532, −8.079304047400714627085982855274, −7.69244698076115102142261904486, −6.22383857184382650748421920789, −5.46717335456348787719099972759, −4.35540056013681339051703945396, −3.58050740531579197913814825165, −2.71692763830758232490228673908, −1.40386554195830410873168783651,
1.40386554195830410873168783651, 2.71692763830758232490228673908, 3.58050740531579197913814825165, 4.35540056013681339051703945396, 5.46717335456348787719099972759, 6.22383857184382650748421920789, 7.69244698076115102142261904486, 8.079304047400714627085982855274, 9.120226310956534660879786635532, 9.529025836474126299504449375420
