| L(s) = 1 | − 1.88·2-s + 1.54·4-s − 3.12·5-s + 2.89·7-s + 0.858·8-s + 5.87·10-s − 11-s + 0.612·13-s − 5.44·14-s − 4.70·16-s + 5.09·17-s + 2.06·19-s − 4.81·20-s + 1.88·22-s − 0.735·23-s + 4.74·25-s − 1.15·26-s + 4.46·28-s − 0.761·29-s − 4.32·31-s + 7.13·32-s − 9.59·34-s − 9.03·35-s − 1.64·37-s − 3.89·38-s − 2.68·40-s − 6.49·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.771·4-s − 1.39·5-s + 1.09·7-s + 0.303·8-s + 1.85·10-s − 0.301·11-s + 0.169·13-s − 1.45·14-s − 1.17·16-s + 1.23·17-s + 0.474·19-s − 1.07·20-s + 0.401·22-s − 0.153·23-s + 0.948·25-s − 0.225·26-s + 0.844·28-s − 0.141·29-s − 0.777·31-s + 1.26·32-s − 1.64·34-s − 1.52·35-s − 0.269·37-s − 0.631·38-s − 0.423·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 13 | \( 1 - 0.612T + 13T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 - 2.06T + 19T^{2} \) |
| 23 | \( 1 + 0.735T + 23T^{2} \) |
| 29 | \( 1 + 0.761T + 29T^{2} \) |
| 31 | \( 1 + 4.32T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 + 9.20T + 43T^{2} \) |
| 47 | \( 1 + 3.57T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 + 2.36T + 71T^{2} \) |
| 73 | \( 1 + 7.36T + 73T^{2} \) |
| 79 | \( 1 + 1.49T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 - 9.67T + 89T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
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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
−7.77843793688882597097259042806, −7.49047771088906060399251585511, −6.75919684599675052328720364178, −5.44079231298881922161716945611, −4.86349529489975266018663171959, −3.96878295260412687662615365507, −3.23516222302649963299022393813, −1.90626658867245593585975265043, −1.07854877141477012459639214717, 0,
1.07854877141477012459639214717, 1.90626658867245593585975265043, 3.23516222302649963299022393813, 3.96878295260412687662615365507, 4.86349529489975266018663171959, 5.44079231298881922161716945611, 6.75919684599675052328720364178, 7.49047771088906060399251585511, 7.77843793688882597097259042806
