| L(s) = 1 | + 1.52e6·2-s − 3.48e9·3-s + 1.18e11·4-s + 2.08e14·5-s − 5.30e15·6-s − 2.54e16·7-s − 3.16e18·8-s + 1.21e19·9-s + 3.17e20·10-s − 6.14e20·11-s − 4.11e20·12-s + 5.75e22·13-s − 3.87e22·14-s − 7.28e23·15-s − 5.08e24·16-s − 1.56e25·17-s + 1.85e25·18-s − 2.61e26·19-s + 2.46e25·20-s + 8.86e25·21-s − 9.35e26·22-s + 2.02e27·23-s + 1.10e28·24-s − 1.85e27·25-s + 8.75e28·26-s − 4.23e28·27-s − 3.00e27·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s − 0.577·3-s + 0.0537·4-s + 0.979·5-s − 0.592·6-s − 0.120·7-s − 0.971·8-s + 0.333·9-s + 1.00·10-s − 0.275·11-s − 0.0310·12-s + 0.839·13-s − 0.123·14-s − 0.565·15-s − 1.05·16-s − 0.932·17-s + 0.342·18-s − 1.59·19-s + 0.0526·20-s + 0.0695·21-s − 0.282·22-s + 0.246·23-s + 0.560·24-s − 0.0407·25-s + 0.861·26-s − 0.192·27-s − 0.00646·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(21)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{43}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3.48e9T \) |
| good | 2 | \( 1 - 1.52e6T + 2.19e12T^{2} \) |
| 5 | \( 1 - 2.08e14T + 4.54e28T^{2} \) |
| 7 | \( 1 + 2.54e16T + 4.45e34T^{2} \) |
| 11 | \( 1 + 6.14e20T + 4.97e42T^{2} \) |
| 13 | \( 1 - 5.75e22T + 4.69e45T^{2} \) |
| 17 | \( 1 + 1.56e25T + 2.80e50T^{2} \) |
| 19 | \( 1 + 2.61e26T + 2.68e52T^{2} \) |
| 23 | \( 1 - 2.02e27T + 6.77e55T^{2} \) |
| 29 | \( 1 + 1.88e30T + 9.08e59T^{2} \) |
| 31 | \( 1 + 1.56e30T + 1.39e61T^{2} \) |
| 37 | \( 1 + 1.62e32T + 1.97e64T^{2} \) |
| 41 | \( 1 + 1.30e32T + 1.33e66T^{2} \) |
| 43 | \( 1 - 5.91e32T + 9.38e66T^{2} \) |
| 47 | \( 1 - 3.41e34T + 3.59e68T^{2} \) |
| 53 | \( 1 - 2.09e35T + 4.95e70T^{2} \) |
| 59 | \( 1 + 8.11e35T + 4.02e72T^{2} \) |
| 61 | \( 1 + 6.81e36T + 1.57e73T^{2} \) |
| 67 | \( 1 - 3.85e37T + 7.39e74T^{2} \) |
| 71 | \( 1 - 3.64e37T + 7.97e75T^{2} \) |
| 73 | \( 1 - 1.66e38T + 2.49e76T^{2} \) |
| 79 | \( 1 - 1.86e38T + 6.34e77T^{2} \) |
| 83 | \( 1 + 3.07e38T + 4.81e78T^{2} \) |
| 89 | \( 1 - 9.41e39T + 8.41e79T^{2} \) |
| 97 | \( 1 + 9.48e40T + 2.86e81T^{2} \) |
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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
−15.28650674178046418145002508098, −13.64323695281511763205586744636, −12.74598318780724631277199282153, −10.88377241235052085853871590569, −9.074149955972002699606695312444, −6.41320767912155713763888713958, −5.42439983694208239816088056900, −3.95380537396362182708300447045, −2.06144714555094979220189806280, 0,
2.06144714555094979220189806280, 3.95380537396362182708300447045, 5.42439983694208239816088056900, 6.41320767912155713763888713958, 9.074149955972002699606695312444, 10.88377241235052085853871590569, 12.74598318780724631277199282153, 13.64323695281511763205586744636, 15.28650674178046418145002508098
