| L(s) = 1 | + (0.0475 + 0.998i)2-s + (−0.297 + 1.70i)3-s + (−0.995 + 0.0950i)4-s + (0.616 + 0.711i)5-s + (−1.71 − 0.216i)6-s + (−1.81 + 3.51i)7-s + (−0.142 − 0.989i)8-s + (−2.82 − 1.01i)9-s + (−0.681 + 0.649i)10-s + (1.32 − 3.82i)11-s + (0.134 − 1.72i)12-s + (−2.16 + 2.75i)13-s + (−3.59 − 1.64i)14-s + (−1.39 + 0.839i)15-s + (0.981 − 0.189i)16-s + (−0.337 + 3.53i)17-s + ⋯ |
| L(s) = 1 | + (0.0336 + 0.706i)2-s + (−0.171 + 0.985i)3-s + (−0.497 + 0.0475i)4-s + (0.275 + 0.318i)5-s + (−0.701 − 0.0883i)6-s + (−0.684 + 1.32i)7-s + (−0.0503 − 0.349i)8-s + (−0.940 − 0.338i)9-s + (−0.215 + 0.205i)10-s + (0.398 − 1.15i)11-s + (0.0387 − 0.498i)12-s + (−0.600 + 0.763i)13-s + (−0.960 − 0.438i)14-s + (−0.360 + 0.216i)15-s + (0.245 − 0.0473i)16-s + (−0.0818 + 0.857i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.139551 - 0.905201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.139551 - 0.905201i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0475 - 0.998i)T \) |
| 3 | \( 1 + (0.297 - 1.70i)T \) |
| 67 | \( 1 + (-4.17 - 7.04i)T \) |
| good | 5 | \( 1 + (-0.616 - 0.711i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.81 - 3.51i)T + (-4.06 - 5.70i)T^{2} \) |
| 11 | \( 1 + (-1.32 + 3.82i)T + (-8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (2.16 - 2.75i)T + (-3.06 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.337 - 3.53i)T + (-16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (-2.52 + 1.29i)T + (11.0 - 15.4i)T^{2} \) |
| 23 | \( 1 + (1.92 + 0.467i)T + (20.4 + 10.5i)T^{2} \) |
| 29 | \( 1 + (6.00 - 3.46i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.85 - 3.63i)T + (-7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (5.64 - 9.77i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 2.35i)T + (-13.4 - 38.7i)T^{2} \) |
| 43 | \( 1 + (-6.65 + 3.04i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-5.23 + 5.49i)T + (-2.23 - 46.9i)T^{2} \) |
| 53 | \( 1 + (4.98 - 10.9i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.10 - 0.158i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-2.40 + 0.830i)T + (47.9 - 37.7i)T^{2} \) |
| 71 | \( 1 + (-0.514 - 5.38i)T + (-69.7 + 13.4i)T^{2} \) |
| 73 | \( 1 + (0.140 + 0.405i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-5.01 - 12.5i)T + (-57.1 + 54.5i)T^{2} \) |
| 83 | \( 1 + (-1.92 - 9.99i)T + (-77.0 + 30.8i)T^{2} \) |
| 89 | \( 1 + (3.68 - 12.5i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-4.67 - 2.69i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.78735066534887721410564796592, −10.71226336048847612213798733191, −9.732773598811958446348887072167, −8.997840657868716465721186337043, −8.427828506745704964584728546688, −6.73628200360440480527185320873, −5.98144381498733439409149874637, −5.26792753988022741513299434277, −3.89729604553182525076881888700, −2.76090793895843175176846875452,
0.59753162700937383825835608891, 2.04507092064006861259564095926, 3.48029228918136949075053004105, 4.79037061129516747132044538156, 5.97069480726384200965020422504, 7.32640315245333452973430204256, 7.59184424009231581531528018174, 9.287612433462424367107137461579, 9.841670263276454398866831641344, 10.84667602433431750459579945482
