| L(s) = 1 | − 1.82·2-s + 1.33·4-s − 5-s + 1.74·7-s + 1.20·8-s + 1.82·10-s − 5.35·13-s − 3.19·14-s − 4.88·16-s − 7.33·17-s + 6.01·19-s − 1.33·20-s − 8.35·23-s + 25-s + 9.78·26-s + 2.33·28-s − 5.53·29-s − 0.0213·31-s + 6.50·32-s + 13.3·34-s − 1.74·35-s − 1.12·37-s − 10.9·38-s − 1.20·40-s + 3.65·41-s − 1.96·43-s + 15.2·46-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.669·4-s − 0.447·5-s + 0.660·7-s + 0.427·8-s + 0.577·10-s − 1.48·13-s − 0.853·14-s − 1.22·16-s − 1.77·17-s + 1.37·19-s − 0.299·20-s − 1.74·23-s + 0.200·25-s + 1.91·26-s + 0.441·28-s − 1.02·29-s − 0.00383·31-s + 1.15·32-s + 2.29·34-s − 0.295·35-s − 0.184·37-s − 1.78·38-s − 0.191·40-s + 0.571·41-s − 0.300·43-s + 2.25·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4459417575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4459417575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 + 7.33T + 17T^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 + 8.35T + 23T^{2} \) |
| 29 | \( 1 + 5.53T + 29T^{2} \) |
| 31 | \( 1 + 0.0213T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 + 1.67T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 - 9.27T + 61T^{2} \) |
| 67 | \( 1 - 0.350T + 67T^{2} \) |
| 71 | \( 1 + 4.53T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 2.56T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 1.89T + 97T^{2} \) |
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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
−8.108895245514226356124484548450, −7.63948035584590490320765833463, −7.15405117234147970819575208702, −6.28007242618763622191049265684, −5.07976331448573435187157462884, −4.62761893519256743056920531612, −3.71129055558268243168103799640, −2.36365885036176747101590122815, −1.79164885240534944179864022575, −0.42584992172214066086248064672,
0.42584992172214066086248064672, 1.79164885240534944179864022575, 2.36365885036176747101590122815, 3.71129055558268243168103799640, 4.62761893519256743056920531612, 5.07976331448573435187157462884, 6.28007242618763622191049265684, 7.15405117234147970819575208702, 7.63948035584590490320765833463, 8.108895245514226356124484548450
