L(s) = 1 | + 0.804·3-s − 1.89·5-s − 4.89·7-s − 2.35·9-s + 11-s − 3.11·13-s − 1.52·15-s + 2·17-s − 19-s − 3.93·21-s − 8.24·23-s − 1.42·25-s − 4.30·27-s + 4.78·29-s + 7.78·31-s + 0.804·33-s + 9.24·35-s + 0.462·37-s − 2.50·39-s + 9.59·41-s − 2.74·43-s + 4.44·45-s + 8.70·47-s + 16.9·49-s + 1.60·51-s + 2·53-s − 1.89·55-s + ⋯ |
L(s) = 1 | + 0.464·3-s − 0.845·5-s − 1.84·7-s − 0.784·9-s + 0.301·11-s − 0.863·13-s − 0.392·15-s + 0.485·17-s − 0.229·19-s − 0.858·21-s − 1.71·23-s − 0.285·25-s − 0.828·27-s + 0.888·29-s + 1.39·31-s + 0.140·33-s + 1.56·35-s + 0.0760·37-s − 0.401·39-s + 1.49·41-s − 0.418·43-s + 0.662·45-s + 1.26·47-s + 2.41·49-s + 0.225·51-s + 0.274·53-s − 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7476107152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7476107152\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 0.804T + 3T^{2} \) |
| 5 | \( 1 + 1.89T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 13 | \( 1 + 3.11T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 - 0.462T + 37T^{2} \) |
| 41 | \( 1 - 9.59T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 5.79T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 + 7.78T + 79T^{2} \) |
| 83 | \( 1 - 6.55T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 7.35T + 97T^{2} \) |
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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
−8.646240964903773263614325097789, −7.82119346377912143013628965011, −7.31341902487334992647379389636, −6.17245935033412757972158283393, −5.99394175010048595847556321425, −4.53619344600921363840534939668, −3.79069595488229119025126157581, −3.06145506609494746425749564675, −2.40170104893698764821493491996, −0.46809132052956985688790627430,
0.46809132052956985688790627430, 2.40170104893698764821493491996, 3.06145506609494746425749564675, 3.79069595488229119025126157581, 4.53619344600921363840534939668, 5.99394175010048595847556321425, 6.17245935033412757972158283393, 7.31341902487334992647379389636, 7.82119346377912143013628965011, 8.646240964903773263614325097789