L(s) = 1 | + 2.60·2-s − 3-s + 4.78·4-s − 3.78·5-s − 2.60·6-s + 7-s + 7.26·8-s + 9-s − 9.86·10-s − 2.55·11-s − 4.78·12-s + 2.60·14-s + 3.78·15-s + 9.34·16-s + 3.21·17-s + 2.60·18-s − 8.44·19-s − 18.1·20-s − 21-s − 6.65·22-s − 3.97·23-s − 7.26·24-s + 9.34·25-s − 27-s + 4.78·28-s − 2.86·29-s + 9.86·30-s + ⋯ |
L(s) = 1 | + 1.84·2-s − 0.577·3-s + 2.39·4-s − 1.69·5-s − 1.06·6-s + 0.377·7-s + 2.56·8-s + 0.333·9-s − 3.11·10-s − 0.770·11-s − 1.38·12-s + 0.696·14-s + 0.977·15-s + 2.33·16-s + 0.778·17-s + 0.614·18-s − 1.93·19-s − 4.05·20-s − 0.218·21-s − 1.41·22-s − 0.829·23-s − 1.48·24-s + 1.86·25-s − 0.192·27-s + 0.904·28-s − 0.532·29-s + 1.80·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 5 | \( 1 + 3.78T + 5T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 + 8.44T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 - 0.868T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 4.10T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 1.23T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
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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
−7.86338455486197756336371674485, −7.32332037725342437365441549577, −6.44942218049928812227011731046, −5.82306410409547164803667739961, −4.82151635523446890151150816245, −4.48125553404344436120790001634, −3.74154319377687297618384346316, −2.98904227630736429206481796971, −1.81227849698730758177160540804, 0,
1.81227849698730758177160540804, 2.98904227630736429206481796971, 3.74154319377687297618384346316, 4.48125553404344436120790001634, 4.82151635523446890151150816245, 5.82306410409547164803667739961, 6.44942218049928812227011731046, 7.32332037725342437365441549577, 7.86338455486197756336371674485