| L(s) = 1 | − 0.962·2-s + 2.32·3-s − 1.07·4-s − 5-s − 2.23·6-s + 2.98·7-s + 2.95·8-s + 2.40·9-s + 0.962·10-s − 11-s − 2.49·12-s − 0.728·13-s − 2.86·14-s − 2.32·15-s − 0.699·16-s + 0.869·17-s − 2.31·18-s − 5.99·19-s + 1.07·20-s + 6.93·21-s + 0.962·22-s − 4.15·23-s + 6.87·24-s + 25-s + 0.700·26-s − 1.37·27-s − 3.20·28-s + ⋯ |
| L(s) = 1 | − 0.680·2-s + 1.34·3-s − 0.536·4-s − 0.447·5-s − 0.913·6-s + 1.12·7-s + 1.04·8-s + 0.802·9-s + 0.304·10-s − 0.301·11-s − 0.720·12-s − 0.201·13-s − 0.766·14-s − 0.600·15-s − 0.174·16-s + 0.211·17-s − 0.546·18-s − 1.37·19-s + 0.240·20-s + 1.51·21-s + 0.205·22-s − 0.867·23-s + 1.40·24-s + 0.200·25-s + 0.137·26-s − 0.265·27-s − 0.604·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 0.962T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 7 | \( 1 - 2.98T + 7T^{2} \) |
| 13 | \( 1 + 0.728T + 13T^{2} \) |
| 17 | \( 1 - 0.869T + 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 + 0.861T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 + 0.518T + 37T^{2} \) |
| 41 | \( 1 - 0.372T + 41T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 - 8.16T + 47T^{2} \) |
| 53 | \( 1 + 0.188T + 53T^{2} \) |
| 59 | \( 1 + 2.73T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 + 1.13T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
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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
−8.305454928041062002498970821942, −7.68819365668755014470056707407, −7.16076821625100977447826976328, −5.79174128221836479790001944115, −4.80048557726439359583725140182, −4.19503317532095998674872516815, −3.44914652747765291330988553971, −2.24173474682610220738692185384, −1.58443301215438857160993879539, 0,
1.58443301215438857160993879539, 2.24173474682610220738692185384, 3.44914652747765291330988553971, 4.19503317532095998674872516815, 4.80048557726439359583725140182, 5.79174128221836479790001944115, 7.16076821625100977447826976328, 7.68819365668755014470056707407, 8.305454928041062002498970821942
