| L(s) = 1 | + (−1.90e4 − 3.30e4i)2-s + (2.39e6 − 4.14e6i)3-s + (−4.60e8 + 7.98e8i)4-s + (9.98e9 + 1.72e10i)5-s − 1.82e11·6-s + (1.32e12 + 1.21e12i)7-s + (1.46e13 + 0.00585i)8-s + (−1.14e13 − 1.98e13i)9-s + (3.81e14 − 6.60e14i)10-s + (3.14e14 − 5.45e14i)11-s + (2.20e15 + 3.81e15i)12-s + 7.96e15·13-s + (1.49e16 − 6.68e16i)14-s + (9.54e16 + 16i)15-s + (−3.31e16 − 5.74e16i)16-s + (1.37e17 − 2.38e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 − 1.42i)2-s + (0.288 − 0.499i)3-s + (−0.858 + 1.48i)4-s + (0.731 + 1.26i)5-s − 0.951·6-s + (0.735 + 0.677i)7-s + 1.18·8-s + (−0.166 − 0.288i)9-s + (1.20 − 2.08i)10-s + (0.249 − 0.432i)11-s + (0.495 + 0.858i)12-s + 0.560·13-s + (0.360 − 1.60i)14-s + 0.844·15-s + (−0.115 − 0.199i)16-s + (0.198 − 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.976514031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976514031\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.39e6 + 4.14e6i)T \) |
| 7 | \( 1 + (-1.32e12 - 1.21e12i)T \) |
| good | 2 | \( 1 + (1.90e4 + 3.30e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-9.98e9 - 1.72e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-3.14e14 + 5.45e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 - 7.96e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-1.37e17 + 2.38e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.60e18 + 4.51e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-3.72e19 - 6.44e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 + 2.01e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-1.58e20 + 2.73e20i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (3.33e22 + 5.76e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 2.44e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 7.99e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (1.13e24 + 1.97e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-2.95e24 + 5.12e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-3.85e24 + 6.67e24i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-4.70e25 - 8.15e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (-1.28e26 + 2.21e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 1.03e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + (5.91e26 - 1.02e27i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-2.56e27 - 4.45e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 9.81e26T + 4.50e55T^{2} \) |
| 89 | \( 1 + (1.19e28 + 2.06e28i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 3.95e28T + 4.13e57T^{2} \) |
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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
−11.32247205173959817451276614391, −10.99909100992518226424331952090, −9.502012622477174065638684991089, −8.671831329664164671789877480301, −7.22224996865507979437371558905, −5.76317791643040094721270967282, −3.58007322645106534674198505883, −2.52375435138455451810507813999, −1.96157531683338003515345030045, −0.77420735064035034538525468974,
0.77080699528682720766401807815, 1.69720791143748272307166980510, 4.17646392729647879081154058334, 5.17481665258617945272832981530, 6.24673402863485006197526005155, 7.78965103222588026857669526169, 8.608504207532644974990013049011, 9.468083075399497697275446039422, 10.59504314706385296042479001987, 12.71373230863275052916351321174
