| L(s) = 1 | + (0.830 − 1.81i)2-s + (−0.858 − 0.252i)3-s + (−2.61 − 3.02i)4-s + (9.50 − 6.10i)5-s + (−1.17 + 1.35i)6-s + (3.59 − 25.0i)7-s + (−7.67 + 2.25i)8-s + (−22.0 − 14.1i)9-s + (−3.21 − 22.3i)10-s + (26.9 + 59.0i)11-s + (1.48 + 3.25i)12-s + (3.47 + 24.1i)13-s + (−42.5 − 27.3i)14-s + (−9.70 + 2.84i)15-s + (−2.27 + 15.8i)16-s + (52.5 − 60.6i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.165 − 0.0485i)3-s + (−0.327 − 0.377i)4-s + (0.850 − 0.546i)5-s + (−0.0797 + 0.0920i)6-s + (0.194 − 1.35i)7-s + (−0.339 + 0.0996i)8-s + (−0.816 − 0.524i)9-s + (−0.101 − 0.707i)10-s + (0.739 + 1.61i)11-s + (0.0357 + 0.0783i)12-s + (0.0740 + 0.515i)13-s + (−0.812 − 0.521i)14-s + (−0.167 + 0.0490i)15-s + (−0.0355 + 0.247i)16-s + (0.750 − 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0483 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0483 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.12446 - 1.07138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12446 - 1.07138i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.830 + 1.81i)T \) |
| 23 | \( 1 + (108. - 18.2i)T \) |
| good | 3 | \( 1 + (0.858 + 0.252i)T + (22.7 + 14.5i)T^{2} \) |
| 5 | \( 1 + (-9.50 + 6.10i)T + (51.9 - 113. i)T^{2} \) |
| 7 | \( 1 + (-3.59 + 25.0i)T + (-329. - 96.6i)T^{2} \) |
| 11 | \( 1 + (-26.9 - 59.0i)T + (-871. + 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-3.47 - 24.1i)T + (-2.10e3 + 618. i)T^{2} \) |
| 17 | \( 1 + (-52.5 + 60.6i)T + (-699. - 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-94.3 - 108. i)T + (-976. + 6.78e3i)T^{2} \) |
| 29 | \( 1 + (23.2 - 26.8i)T + (-3.47e3 - 2.41e4i)T^{2} \) |
| 31 | \( 1 + (70.4 - 20.6i)T + (2.50e4 - 1.61e4i)T^{2} \) |
| 37 | \( 1 + (186. + 119. i)T + (2.10e4 + 4.60e4i)T^{2} \) |
| 41 | \( 1 + (-251. + 161. i)T + (2.86e4 - 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-285. - 83.9i)T + (6.68e4 + 4.29e4i)T^{2} \) |
| 47 | \( 1 + 57.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-74.0 + 515. i)T + (-1.42e5 - 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-9.90 - 68.8i)T + (-1.97e5 + 5.78e4i)T^{2} \) |
| 61 | \( 1 + (332. - 97.7i)T + (1.90e5 - 1.22e5i)T^{2} \) |
| 67 | \( 1 + (359. - 787. i)T + (-1.96e5 - 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-31.1 + 68.1i)T + (-2.34e5 - 2.70e5i)T^{2} \) |
| 73 | \( 1 + (-206. - 238. i)T + (-5.53e4 + 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-37.6 - 261. i)T + (-4.73e5 + 1.38e5i)T^{2} \) |
| 83 | \( 1 + (682. + 438. i)T + (2.37e5 + 5.20e5i)T^{2} \) |
| 89 | \( 1 + (1.46e3 + 430. i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (29.3 - 18.8i)T + (3.79e5 - 8.30e5i)T^{2} \) |
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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
−14.32835237329286144779390757132, −14.04986495825326846697721046110, −12.53733149463701427674230484375, −11.66338710711759451222473721301, −10.07176232931607928296228518285, −9.368605803163532253533660455451, −7.29191069615387094112626997182, −5.55355244838324092043558383249, −3.98234420803563895990390939032, −1.44468645503051984519596473880,
2.91632662550928744602343707650, 5.60265876969084621224781445292, 6.05841500494165779833429895221, 8.164261831801113669910456498053, 9.218231665717857570879411296383, 10.94821805803117499832187435232, 12.07486497675324227783521257674, 13.70021827651385511658363812217, 14.29485100675946647316266073095, 15.51087089209070843541287951602
