| L(s) = 1 | − 2.81·3-s + 0.334·5-s + 4.24·7-s + 4.90·9-s − 0.915·11-s + 4.81·13-s − 0.941·15-s − 5.38·17-s + 4.34·19-s − 11.9·21-s − 8.10·23-s − 4.88·25-s − 5.34·27-s − 6.45·29-s − 10.7·31-s + 2.57·33-s + 1.42·35-s + 5.16·37-s − 13.5·39-s − 0.612·41-s + 1.85·43-s + 1.64·45-s + 2.21·47-s + 11.0·49-s + 15.1·51-s + 0.00846·53-s − 0.306·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.149·5-s + 1.60·7-s + 1.63·9-s − 0.276·11-s + 1.33·13-s − 0.243·15-s − 1.30·17-s + 0.997·19-s − 2.60·21-s − 1.69·23-s − 0.977·25-s − 1.02·27-s − 1.19·29-s − 1.93·31-s + 0.447·33-s + 0.240·35-s + 0.849·37-s − 2.16·39-s − 0.0956·41-s + 0.282·43-s + 0.244·45-s + 0.322·47-s + 1.57·49-s + 2.12·51-s + 0.00116·53-s − 0.0413·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 5 | \( 1 - 0.334T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 0.915T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 8.10T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 + 0.612T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 - 2.21T + 47T^{2} \) |
| 53 | \( 1 - 0.00846T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 + 1.48T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 17.6T + 79T^{2} \) |
| 83 | \( 1 + 7.45T + 83T^{2} \) |
| 89 | \( 1 + 0.520T + 89T^{2} \) |
| 97 | \( 1 + 6.33T + 97T^{2} \) |
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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
−7.895325596367150585223223108714, −7.44353379529267457294594005549, −6.39189443228953351469979201957, −5.70825540307370013006959042325, −5.37496588747992115839702581049, −4.39923215358971010122049559136, −3.86061737139884758807990003701, −2.03072915276865407080076258786, −1.38390311239823449143319902628, 0,
1.38390311239823449143319902628, 2.03072915276865407080076258786, 3.86061737139884758807990003701, 4.39923215358971010122049559136, 5.37496588747992115839702581049, 5.70825540307370013006959042325, 6.39189443228953351469979201957, 7.44353379529267457294594005549, 7.895325596367150585223223108714
