L(s) = 1 | + (1.83 − 3.18i)5-s + (−1.55 + 2.14i)7-s + (0.301 + 0.522i)11-s + (2.62 + 4.55i)13-s + (2.12 − 3.68i)17-s + (3.68 + 6.38i)19-s + (0.578 − 1.00i)23-s + (−4.25 − 7.37i)25-s + (−3.98 + 6.90i)29-s + 3.15·31-s + (3.95 + 8.88i)35-s + (0.00266 + 0.00462i)37-s + (−2.00 − 3.48i)41-s + (−3.66 + 6.34i)43-s + 12.2·47-s + ⋯ |
L(s) = 1 | + (0.822 − 1.42i)5-s + (−0.588 + 0.808i)7-s + (0.0909 + 0.157i)11-s + (0.729 + 1.26i)13-s + (0.515 − 0.892i)17-s + (0.845 + 1.46i)19-s + (0.120 − 0.209i)23-s + (−0.851 − 1.47i)25-s + (−0.740 + 1.28i)29-s + 0.565·31-s + (0.668 + 1.50i)35-s + (0.000438 + 0.000760i)37-s + (−0.313 − 0.543i)41-s + (−0.558 + 0.967i)43-s + 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.056934060\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.056934060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.55 - 2.14i)T \) |
good | 5 | \( 1 + (-1.83 + 3.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.301 - 0.522i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.62 - 4.55i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.12 + 3.68i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.68 - 6.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.578 + 1.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.98 - 6.90i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 + (-0.00266 - 0.00462i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.00 + 3.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.66 - 6.34i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + (4.64 - 8.05i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 6.61T + 59T^{2} \) |
| 61 | \( 1 + 1.93T + 61T^{2} \) |
| 67 | \( 1 - 8.63T + 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 + (5.33 - 9.24i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 + (-6.24 + 10.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.09 + 7.09i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.77 + 11.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.032621628608571576597580185791, −8.628708719823718413023467027254, −7.55876231361564499374396346541, −6.53595774941463569906782021206, −5.74061522809835396535779969350, −5.26782931243604569133767173495, −4.30638697689798703629063470954, −3.25241256229842339814055600340, −1.97084096274729012896539998535, −1.15706531606522186851967019363,
0.832045843684891462374552495561, 2.34184721464260371445664396610, 3.24278308681371881729249443225, 3.76672820760534319735927282140, 5.25126921006018400465238614715, 6.04312758190009438902923361588, 6.60123537637285141268596128017, 7.36288194319183202469912571970, 8.050693294844298023456131372772, 9.198709784416230604836763848400