| L(s) = 1 | − 4.66·2-s − 3·3-s + 13.7·4-s + 4.88·5-s + 14.0·6-s + 26.8·7-s − 27.0·8-s + 9·9-s − 22.8·10-s + 9.83·11-s − 41.3·12-s − 23.9·13-s − 125.·14-s − 14.6·15-s + 15.9·16-s + 33.1·17-s − 42.0·18-s + 67.4·20-s − 80.4·21-s − 45.9·22-s + 128.·23-s + 81.2·24-s − 101.·25-s + 112.·26-s − 27·27-s + 370.·28-s + 263.·29-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 0.577·3-s + 1.72·4-s + 0.437·5-s + 0.953·6-s + 1.44·7-s − 1.19·8-s + 0.333·9-s − 0.721·10-s + 0.269·11-s − 0.995·12-s − 0.511·13-s − 2.39·14-s − 0.252·15-s + 0.249·16-s + 0.473·17-s − 0.550·18-s + 0.754·20-s − 0.836·21-s − 0.444·22-s + 1.16·23-s + 0.690·24-s − 0.808·25-s + 0.844·26-s − 0.192·27-s + 2.49·28-s + 1.68·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.089220134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089220134\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 4.66T + 8T^{2} \) |
| 5 | \( 1 - 4.88T + 125T^{2} \) |
| 7 | \( 1 - 26.8T + 343T^{2} \) |
| 11 | \( 1 - 9.83T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 220.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 466.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 633.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 142.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 709.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 712.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 323.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 339.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 63.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 275.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 258.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 486.T + 9.12e5T^{2} \) |
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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
−9.581050789769284579215128379773, −8.598149918727765792282908029928, −8.053831643033676241712849587294, −7.21637028640776856870329969186, −6.43366938890497970368989889644, −5.29303942840937656439845124703, −4.48004093274767497047856774448, −2.60242905750274127496891528097, −1.53216634122616596596342443326, −0.795963005155086474626069675874,
0.795963005155086474626069675874, 1.53216634122616596596342443326, 2.60242905750274127496891528097, 4.48004093274767497047856774448, 5.29303942840937656439845124703, 6.43366938890497970368989889644, 7.21637028640776856870329969186, 8.053831643033676241712849587294, 8.598149918727765792282908029928, 9.581050789769284579215128379773
