L(s) = 1 | + (2.24 + 0.602i)2-s + (−0.258 − 0.965i)3-s + (2.95 + 1.70i)4-s − 2.32i·6-s + (2.59 + 0.519i)7-s + (2.33 + 2.33i)8-s + (−0.866 + 0.499i)9-s + (−1.76 + 3.05i)11-s + (0.884 − 3.30i)12-s + (4.49 − 4.49i)13-s + (5.51 + 2.73i)14-s + (0.421 + 0.729i)16-s + (−1.79 + 0.481i)17-s + (−2.24 + 0.602i)18-s + (0.0699 + 0.121i)19-s + ⋯ |
L(s) = 1 | + (1.58 + 0.425i)2-s + (−0.149 − 0.557i)3-s + (1.47 + 0.854i)4-s − 0.950i·6-s + (0.980 + 0.196i)7-s + (0.824 + 0.824i)8-s + (−0.288 + 0.166i)9-s + (−0.531 + 0.921i)11-s + (0.255 − 0.952i)12-s + (1.24 − 1.24i)13-s + (1.47 + 0.730i)14-s + (0.105 + 0.182i)16-s + (−0.436 + 0.116i)17-s + (−0.529 + 0.141i)18-s + (0.0160 + 0.0277i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.40880 + 0.377779i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.40880 + 0.377779i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.59 - 0.519i)T \) |
good | 2 | \( 1 + (-2.24 - 0.602i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (1.76 - 3.05i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.49 + 4.49i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.79 - 0.481i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.0699 - 0.121i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.997 - 3.72i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.01iT - 29T^{2} \) |
| 31 | \( 1 + (4.56 + 2.63i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.61 + 1.50i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 0.903iT - 41T^{2} \) |
| 43 | \( 1 + (2.38 + 2.38i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.639 + 2.38i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.71 - 0.726i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.15 - 5.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.69 - 5.01i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.77 - 10.3i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 5.09T + 71T^{2} \) |
| 73 | \( 1 + (2.42 + 9.04i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (7.30 - 4.21i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.37 + 7.37i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.75 + 3.03i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.70 + 8.70i)T + 97iT^{2} \) |
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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
−11.23237050781915618065343197717, −10.41834053222081579068049638031, −8.796311775538684485832713887614, −7.77304093422078551622648403893, −7.15537231834026244240873509759, −5.91081138691055004025754165857, −5.39606894111916872260936684728, −4.39914133016634969280564832328, −3.20885839200609436081409404294, −1.85173727240845257915398700529,
1.83275414716855389790175032761, 3.26994842435677556509027046141, 4.19409492574861971114613432464, 4.92154191331864789142425126256, 5.87085318312192755932377030661, 6.70632211548233231062657514482, 8.235919622481819492175685198233, 9.033134473640886024737209888230, 10.57290997924961370341122802747, 11.08328002112905626581339710456