| L(s) = 1 | + (1.91 + 0.563i)2-s + (4.34 − 5.01i)3-s + (3.36 + 2.16i)4-s + (−1.02 + 7.16i)5-s + (11.1 − 7.17i)6-s + (−7.28 − 15.9i)7-s + (5.23 + 6.04i)8-s + (−2.42 − 16.8i)9-s + (−6.01 + 13.1i)10-s + (13.8 − 4.07i)11-s + (25.4 − 7.47i)12-s + (−34.8 + 76.4i)13-s + (−4.98 − 34.6i)14-s + (31.4 + 36.2i)15-s + (6.64 + 14.5i)16-s + (−3.64 + 2.33i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (0.836 − 0.965i)3-s + (0.420 + 0.270i)4-s + (−0.0921 + 0.640i)5-s + (0.759 − 0.488i)6-s + (−0.393 − 0.860i)7-s + (0.231 + 0.267i)8-s + (−0.0898 − 0.625i)9-s + (−0.190 + 0.416i)10-s + (0.380 − 0.111i)11-s + (0.612 − 0.179i)12-s + (−0.744 + 1.63i)13-s + (−0.0952 − 0.662i)14-s + (0.541 + 0.624i)15-s + (0.103 + 0.227i)16-s + (−0.0519 + 0.0333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.20752 - 0.310707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20752 - 0.310707i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.91 - 0.563i)T \) |
| 23 | \( 1 + (82.1 + 73.6i)T \) |
| good | 3 | \( 1 + (-4.34 + 5.01i)T + (-3.84 - 26.7i)T^{2} \) |
| 5 | \( 1 + (1.02 - 7.16i)T + (-119. - 35.2i)T^{2} \) |
| 7 | \( 1 + (7.28 + 15.9i)T + (-224. + 259. i)T^{2} \) |
| 11 | \( 1 + (-13.8 + 4.07i)T + (1.11e3 - 719. i)T^{2} \) |
| 13 | \( 1 + (34.8 - 76.4i)T + (-1.43e3 - 1.66e3i)T^{2} \) |
| 17 | \( 1 + (3.64 - 2.33i)T + (2.04e3 - 4.46e3i)T^{2} \) |
| 19 | \( 1 + (92.4 + 59.4i)T + (2.84e3 + 6.23e3i)T^{2} \) |
| 29 | \( 1 + (35.6 - 22.9i)T + (1.01e4 - 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-65.1 - 75.1i)T + (-4.23e3 + 2.94e4i)T^{2} \) |
| 37 | \( 1 + (11.2 + 78.5i)T + (-4.86e4 + 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-38.8 + 270. i)T + (-6.61e4 - 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-112. + 129. i)T + (-1.13e4 - 7.86e4i)T^{2} \) |
| 47 | \( 1 - 530.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (253. + 553. i)T + (-9.74e4 + 1.12e5i)T^{2} \) |
| 59 | \( 1 + (56.1 - 122. i)T + (-1.34e5 - 1.55e5i)T^{2} \) |
| 61 | \( 1 + (-499. - 576. i)T + (-3.23e4 + 2.24e5i)T^{2} \) |
| 67 | \( 1 + (-409. - 120. i)T + (2.53e5 + 1.62e5i)T^{2} \) |
| 71 | \( 1 + (1.00e3 + 294. i)T + (3.01e5 + 1.93e5i)T^{2} \) |
| 73 | \( 1 + (562. + 361. i)T + (1.61e5 + 3.53e5i)T^{2} \) |
| 79 | \( 1 + (539. - 1.18e3i)T + (-3.22e5 - 3.72e5i)T^{2} \) |
| 83 | \( 1 + (27.2 + 189. i)T + (-5.48e5 + 1.61e5i)T^{2} \) |
| 89 | \( 1 + (-574. + 662. i)T + (-1.00e5 - 6.97e5i)T^{2} \) |
| 97 | \( 1 + (182. - 1.27e3i)T + (-8.75e5 - 2.57e5i)T^{2} \) |
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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
−14.64913924137072348256735330648, −14.12961557283323467070049600878, −13.20851118618442463187093251401, −12.06726473188067830010848597177, −10.62252407615467753346637819172, −8.817408168292370254187174614604, −7.18093524143810346042066043782, −6.72289709125992260272984601191, −4.12407124169007427623692392665, −2.34708888164742496070500294538,
2.85024498518751371016211757416, 4.34926626379350770406126341022, 5.81593376226164433515040038426, 8.129016938222932651725834691336, 9.358635793936612927186389782672, 10.37017068406229641851679942676, 12.15026921487019092467643969360, 12.92851058175325410958409039362, 14.45893342996845400714189510280, 15.23290289054704598590954679978
