L(s) = 1 | + (1.42 + 1.13i)2-s + (0.295 + 1.29i)4-s + (−0.0720 − 0.0346i)5-s + (2.51 − 0.832i)7-s + (0.533 − 1.10i)8-s + (−0.0632 − 0.131i)10-s + (−0.0489 − 0.0390i)11-s + (3.06 + 2.44i)13-s + (4.52 + 1.66i)14-s + (4.40 − 2.12i)16-s + (−0.399 + 1.75i)17-s + 5.46i·19-s + (0.0235 − 0.103i)20-s + (−0.0254 − 0.111i)22-s + (−4.14 + 0.945i)23-s + ⋯ |
L(s) = 1 | + (1.00 + 0.804i)2-s + (0.147 + 0.646i)4-s + (−0.0322 − 0.0155i)5-s + (0.949 − 0.314i)7-s + (0.188 − 0.391i)8-s + (−0.0200 − 0.0415i)10-s + (−0.0147 − 0.0117i)11-s + (0.850 + 0.678i)13-s + (1.20 + 0.446i)14-s + (1.10 − 0.530i)16-s + (−0.0969 + 0.424i)17-s + 1.25i·19-s + (0.00527 − 0.0231i)20-s + (−0.00541 − 0.0237i)22-s + (−0.864 + 0.197i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.34063 + 0.981030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.34063 + 0.981030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.51 + 0.832i)T \) |
good | 2 | \( 1 + (-1.42 - 1.13i)T + (0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (0.0720 + 0.0346i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (0.0489 + 0.0390i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-3.06 - 2.44i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.399 - 1.75i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 - 5.46iT - 19T^{2} \) |
| 23 | \( 1 + (4.14 - 0.945i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (8.80 + 2.01i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + 6.38iT - 31T^{2} \) |
| 37 | \( 1 + (-0.385 + 1.68i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-6.08 - 2.92i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.79 + 0.866i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (2.99 - 3.76i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.0985 + 0.0224i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (3.52 - 1.69i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (3.20 + 0.731i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + (2.96 - 0.677i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.291 + 0.232i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + (10.5 + 13.2i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.24 - 4.07i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 - 13.6iT - 97T^{2} \) |
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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
−11.38779035526838962093200035267, −10.44980679313977470745845513969, −9.402790861160203784680455138858, −8.048349990866964940497210053132, −7.56251354977338490269763064846, −6.18759511352455439478720455488, −5.72844916599402127698826668594, −4.32398732212343332718749990099, −3.88451350270801563665287154630, −1.71959164625278906724499572415,
1.69513927521817392085022488308, 2.94929802367718366280401760321, 4.05878442352055617961457884663, 5.07975993591908140186501705231, 5.81467335901007124593484208236, 7.36979160728432185133582578169, 8.293268733390009096726118723436, 9.234118619794806525864086015315, 10.65274086532808393892374460476, 11.15895542935799063001793403267