| L(s) = 1 | + (1.91 − 1.30i)2-s + (−1.70 + 0.303i)3-s + (1.23 − 3.15i)4-s + (0.729 + 3.19i)5-s + (−2.87 + 2.81i)6-s + (−0.499 + 2.59i)7-s + (−0.716 − 3.13i)8-s + (2.81 − 1.03i)9-s + (5.57 + 5.17i)10-s + (0.176 + 0.0851i)11-s + (−1.15 + 5.75i)12-s + (3.53 − 2.41i)13-s + (2.43 + 5.63i)14-s + (−2.21 − 5.22i)15-s + (−0.509 − 0.472i)16-s + (2.60 + 6.63i)17-s + ⋯ |
| L(s) = 1 | + (1.35 − 0.924i)2-s + (−0.984 + 0.175i)3-s + (0.618 − 1.57i)4-s + (0.326 + 1.42i)5-s + (−1.17 + 1.14i)6-s + (−0.188 + 0.982i)7-s + (−0.253 − 1.10i)8-s + (0.938 − 0.345i)9-s + (1.76 + 1.63i)10-s + (0.0533 + 0.0256i)11-s + (−0.332 + 1.66i)12-s + (0.981 − 0.669i)13-s + (0.651 + 1.50i)14-s + (−0.571 − 1.34i)15-s + (−0.127 − 0.118i)16-s + (0.631 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.24909 - 0.0146106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24909 - 0.0146106i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.70 - 0.303i)T \) |
| 7 | \( 1 + (0.499 - 2.59i)T \) |
| good | 2 | \( 1 + (-1.91 + 1.30i)T + (0.730 - 1.86i)T^{2} \) |
| 5 | \( 1 + (-0.729 - 3.19i)T + (-4.50 + 2.16i)T^{2} \) |
| 11 | \( 1 + (-0.176 - 0.0851i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-3.53 + 2.41i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-2.60 - 6.63i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (2.09 + 3.63i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.11 + 5.15i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.882 + 0.133i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (-1.47 - 2.55i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.24 + 0.338i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (6.60 + 2.03i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (-2.94 + 0.907i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (-6.55 + 4.47i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-12.7 + 1.91i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (5.80 - 1.78i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (2.94 + 7.50i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (5.51 + 9.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.88 - 3.62i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.466 - 6.22i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (2.55 - 4.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.03 + 0.706i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (2.99 + 2.03i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (7.35 + 12.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.11301086091335331069731942328, −10.56901939518535447315602118439, −10.12100929982566229999303938241, −8.471915518186048811736296986530, −6.71462971336224951270191633437, −6.04674336338257382177372004343, −5.50311902365731532563155334631, −4.09116539483997146663890830174, −3.16126849710451093955601797034, −1.98183569795628836245830783191,
1.20174522059461816444745449712, 3.84767615632572709566665811121, 4.53446951656601591899301495368, 5.45727702029271296074212168062, 6.09089061151427460686485532022, 7.12608897517097632307905198916, 7.907554288111991865837036572961, 9.308182926883142070656594703411, 10.30477212538545270932428447853, 11.70604207579644046741037555559
