| L(s) = 1 | − 0.445·3-s + 5-s + 0.445·7-s − 2.80·9-s − 3.24·11-s − 0.445·15-s + 2.93·17-s + 3.04·19-s − 0.198·21-s − 2.80·23-s + 25-s + 2.58·27-s + 2.15·29-s − 0.939·31-s + 1.44·33-s + 0.445·35-s − 3·37-s − 4.58·41-s + 4.63·43-s − 2.80·45-s + 7.39·47-s − 6.80·49-s − 1.30·51-s − 13.0·53-s − 3.24·55-s − 1.35·57-s + 3.17·59-s + ⋯ |
| L(s) = 1 | − 0.256·3-s + 0.447·5-s + 0.168·7-s − 0.933·9-s − 0.979·11-s − 0.114·15-s + 0.712·17-s + 0.699·19-s − 0.0432·21-s − 0.584·23-s + 0.200·25-s + 0.496·27-s + 0.400·29-s − 0.168·31-s + 0.251·33-s + 0.0752·35-s − 0.493·37-s − 0.715·41-s + 0.706·43-s − 0.417·45-s + 1.07·47-s − 0.971·49-s − 0.183·51-s − 1.79·53-s − 0.437·55-s − 0.179·57-s + 0.413·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.445T + 3T^{2} \) |
| 7 | \( 1 - 0.445T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 - 2.15T + 29T^{2} \) |
| 31 | \( 1 + 0.939T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 - 0.317T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 - 9.62T + 83T^{2} \) |
| 89 | \( 1 + 5.04T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.120932380730899726464100600824, −7.67136566676092620644416657765, −6.65464932004131972100472104488, −5.78427838099588684973266631367, −5.38756689776330052663480407378, −4.55154015160109677187183609995, −3.28169615236916236210787198856, −2.64905121103709793669959372772, −1.45593528063494355925412779508, 0,
1.45593528063494355925412779508, 2.64905121103709793669959372772, 3.28169615236916236210787198856, 4.55154015160109677187183609995, 5.38756689776330052663480407378, 5.78427838099588684973266631367, 6.65464932004131972100472104488, 7.67136566676092620644416657765, 8.120932380730899726464100600824
