| L(s) = 1 | + (1.91 − 0.565i)2-s + (−2.69 + 1.31i)3-s + (3.35 − 2.17i)4-s + (−1.66 + 9.41i)5-s + (−4.43 + 4.04i)6-s + (4.14 + 4.94i)7-s + (5.21 − 6.06i)8-s + (5.55 − 7.08i)9-s + (2.14 + 18.9i)10-s + (−9.17 + 1.61i)11-s + (−6.21 + 10.2i)12-s + (13.9 − 5.08i)13-s + (10.7 + 7.13i)14-s + (−7.88 − 27.5i)15-s + (6.57 − 14.5i)16-s + (2.55 − 4.43i)17-s + ⋯ |
| L(s) = 1 | + (0.959 − 0.282i)2-s + (−0.899 + 0.437i)3-s + (0.839 − 0.542i)4-s + (−0.332 + 1.88i)5-s + (−0.738 + 0.674i)6-s + (0.592 + 0.705i)7-s + (0.652 − 0.758i)8-s + (0.616 − 0.787i)9-s + (0.214 + 1.89i)10-s + (−0.833 + 0.147i)11-s + (−0.517 + 0.855i)12-s + (1.07 − 0.390i)13-s + (0.767 + 0.509i)14-s + (−0.525 − 1.83i)15-s + (0.411 − 0.911i)16-s + (0.150 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.59476 + 0.813903i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59476 + 0.813903i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.91 + 0.565i)T \) |
| 3 | \( 1 + (2.69 - 1.31i)T \) |
| good | 5 | \( 1 + (1.66 - 9.41i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.14 - 4.94i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (9.17 - 1.61i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-13.9 + 5.08i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-2.55 + 4.43i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (10.2 - 5.89i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.496 - 0.591i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-11.5 - 4.21i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-34.5 + 41.1i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-11.8 + 20.5i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-14.0 + 5.12i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-37.1 + 6.55i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-8.14 - 9.70i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 44.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-24.6 - 4.33i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (50.4 - 42.3i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (0.545 + 1.49i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (6.96 + 4.02i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (11.0 + 19.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (35.5 - 97.6i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (3.74 - 10.2i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-59.2 - 102. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (6.45 + 36.5i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.69670725885257032966279605050, −12.34557168954967727808676595612, −11.33716140329422493652958653896, −10.86873208690297023266399085940, −10.04288033284048102855896109506, −7.72017078630471513552603892629, −6.42917128613105009687453382892, −5.64578729719376165085936609198, −4.06140166514565843412398703255, −2.66559855796660585011418392500,
1.28910084124372316619551338421, 4.32261120411247860758607705449, 4.99325470073320032627932597903, 6.18226822027870565845628335290, 7.71185858415394373166789267597, 8.488641182149154634764075578736, 10.61522895362770331498798844674, 11.56166022184903767266977175225, 12.47199797906557324898309594086, 13.17768179593641757853031253696
