| L(s) = 1 | + (0.340 − 1.26i)2-s + (0.224 + 1.71i)3-s + (0.236 + 0.136i)4-s + (−1.25 + 1.85i)5-s + (2.25 + 0.299i)6-s + (2.32 − 1.25i)7-s + (2.11 − 2.11i)8-s + (−2.89 + 0.769i)9-s + (1.92 + 2.21i)10-s + (−3.38 − 1.95i)11-s + (−0.181 + 0.436i)12-s + (−1.56 − 1.56i)13-s + (−0.807 − 3.38i)14-s + (−3.46 − 1.73i)15-s + (−1.69 − 2.92i)16-s + (2.58 − 0.693i)17-s + ⋯ |
| L(s) = 1 | + (0.240 − 0.897i)2-s + (0.129 + 0.991i)3-s + (0.118 + 0.0681i)4-s + (−0.559 + 0.829i)5-s + (0.921 + 0.122i)6-s + (0.879 − 0.476i)7-s + (0.746 − 0.746i)8-s + (−0.966 + 0.256i)9-s + (0.609 + 0.701i)10-s + (−1.01 − 0.588i)11-s + (−0.0523 + 0.125i)12-s + (−0.434 − 0.434i)13-s + (−0.215 − 0.903i)14-s + (−0.894 − 0.447i)15-s + (−0.422 − 0.731i)16-s + (0.627 − 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22657 - 0.0373146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22657 - 0.0373146i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.224 - 1.71i)T \) |
| 5 | \( 1 + (1.25 - 1.85i)T \) |
| 7 | \( 1 + (-2.32 + 1.25i)T \) |
| good | 2 | \( 1 + (-0.340 + 1.26i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.56 + 1.56i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.58 + 0.693i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.61 - 0.930i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.38 - 0.638i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 0.513T + 29T^{2} \) |
| 31 | \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.60 + 1.77i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 0.308iT - 41T^{2} \) |
| 43 | \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.36 + 5.10i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.498 + 1.85i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.259 - 0.448i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.34 - 8.74i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 15.3iT - 71T^{2} \) |
| 73 | \( 1 + (-2.79 + 0.749i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.37 + 2.52i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.16 - 9.16i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.67 - 9.82i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.81 - 6.81i)T - 97iT^{2} \) |
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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
−13.86960468559094906686018170345, −12.46099271854131731270015475483, −11.27052704194844822334282972088, −10.77880840524231199925538961088, −10.09072057733748490440179601708, −8.267064887130154162109985129140, −7.29148039382786147226254168483, −5.20345071290482051891539274991, −3.81941783381491378346168396626, −2.77732612026732877221391517755,
2.03146238300716129021761020825, 4.81715121360924810281503101819, 5.75470820309427466431828730814, 7.35151424084147504762270334992, 7.85328612111026418827327509830, 8.926251483029510006205350952309, 10.90583600566160765459394741909, 11.95525509040646853269615438380, 12.77813787151423467880683774779, 13.91467102477436123553490575835
