| L(s) = 1 | + (1.79 − 2.40i)3-s + 10.2i·7-s + (−2.53 − 8.63i)9-s − 8.19i·11-s + 13.5i·13-s − 15.4·17-s − 25.4·19-s + (24.5 + 18.3i)21-s + 17.9·23-s + (−25.2 − 9.44i)27-s + 42.0i·29-s − 38.4·31-s + (−19.6 − 14.7i)33-s + 11.8i·37-s + (32.6 + 24.4i)39-s + ⋯ |
| L(s) = 1 | + (0.599 − 0.800i)3-s + 1.45i·7-s + (−0.281 − 0.959i)9-s − 0.744i·11-s + 1.04i·13-s − 0.910·17-s − 1.34·19-s + (1.16 + 0.874i)21-s + 0.778·23-s + (−0.936 − 0.349i)27-s + 1.44i·29-s − 1.24·31-s + (−0.596 − 0.446i)33-s + 0.319i·37-s + (0.836 + 0.626i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8623956015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8623956015\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.79 + 2.40i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10.2iT - 49T^{2} \) |
| 11 | \( 1 + 8.19iT - 121T^{2} \) |
| 13 | \( 1 - 13.5iT - 169T^{2} \) |
| 17 | \( 1 + 15.4T + 289T^{2} \) |
| 19 | \( 1 + 25.4T + 361T^{2} \) |
| 23 | \( 1 - 17.9T + 529T^{2} \) |
| 29 | \( 1 - 42.0iT - 841T^{2} \) |
| 31 | \( 1 + 38.4T + 961T^{2} \) |
| 37 | \( 1 - 11.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 54.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 43.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 82.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 45.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 68.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 44.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 11.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 144.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 63.9iT - 9.40e3T^{2} \) |
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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
−9.336659889022514432050075875189, −8.829351581520122845881586836837, −8.535274963770761287041744096514, −7.29961873033761496998452087292, −6.49564133652434354421108525602, −5.87134480021308929646359158767, −4.69851385530571402177412412776, −3.41101410131779563883220415977, −2.44888652505131439137212797648, −1.63805284239829303282791350157,
0.21447149708571841218663622904, 1.95925465608452610749554549451, 3.14286848028134974517020264198, 4.21886192837647121402417485006, 4.56846323180906975670958668710, 5.85070952512061667393678138944, 7.04765881080402007859560236693, 7.67132626305157747729992213846, 8.505202052327294265658345905851, 9.399414883644660966361445342462
