| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 4.92i·5-s − 5.13i·7-s − 2.82·8-s − 6.96i·10-s − 7.71i·11-s + 0.944·13-s + 7.26i·14-s + 4.00·16-s − 27.6i·17-s + 9.08i·19-s + 9.84i·20-s + 10.9i·22-s + (20.0 + 11.2i)23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.984i·5-s − 0.734i·7-s − 0.353·8-s − 0.696i·10-s − 0.701i·11-s + 0.0726·13-s + 0.519i·14-s + 0.250·16-s − 1.62i·17-s + 0.477i·19-s + 0.492i·20-s + 0.496i·22-s + (0.872 + 0.488i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13089 - 0.295223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13089 - 0.295223i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-20.0 - 11.2i)T \) |
| good | 5 | \( 1 - 4.92iT - 25T^{2} \) |
| 7 | \( 1 + 5.13iT - 49T^{2} \) |
| 11 | \( 1 + 7.71iT - 121T^{2} \) |
| 13 | \( 1 - 0.944T + 169T^{2} \) |
| 17 | \( 1 + 27.6iT - 289T^{2} \) |
| 19 | \( 1 - 9.08iT - 361T^{2} \) |
| 29 | \( 1 - 14.4T + 841T^{2} \) |
| 31 | \( 1 + 0.830T + 961T^{2} \) |
| 37 | \( 1 + 23.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 26.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 92.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 47.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 121. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 8.10iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 23.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 98.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 70.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 50.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 87.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 36.7iT - 9.40e3T^{2} \) |
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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
−11.03056859148987456509151911906, −10.02751019586931179413192914090, −9.254186380753434446792708977803, −8.089335843026234293388896373772, −7.18976684627291127798420084381, −6.58795142038725430441154592226, −5.25950322774663156722765615892, −3.63113612908461775154109433883, −2.62023018096792837983494033703, −0.76348638188769083469959916896,
1.17364445297236135583088092785, 2.51679196852960860579169542406, 4.24656236989896791309170538384, 5.36678402511995097791572597472, 6.41224606665031515172280639590, 7.56666318686016197163638569130, 8.731807927746953789720335011057, 8.900278060438831012688483130101, 10.11014543639750744894994724487, 10.91115039256778542922836044648
