L(s) = 1 | − 3.52·2-s + 1.73i·3-s + 8.39·4-s − 2.23i·5-s − 6.09i·6-s − 15.4·8-s − 2.99·9-s + 7.87i·10-s + 2.59·11-s + 14.5i·12-s − 11.5i·13-s + 3.87·15-s + 20.8·16-s − 23.2i·17-s + 10.5·18-s + 29.9i·19-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 0.577i·3-s + 2.09·4-s − 0.447i·5-s − 1.01i·6-s − 1.93·8-s − 0.333·9-s + 0.787i·10-s + 0.235·11-s + 1.21i·12-s − 0.890i·13-s + 0.258·15-s + 1.30·16-s − 1.36i·17-s + 0.586·18-s + 1.57i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2746134984\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2746134984\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 3.52T + 4T^{2} \) |
| 11 | \( 1 - 2.59T + 121T^{2} \) |
| 13 | \( 1 + 11.5iT - 169T^{2} \) |
| 17 | \( 1 + 23.2iT - 289T^{2} \) |
| 19 | \( 1 - 29.9iT - 361T^{2} \) |
| 23 | \( 1 + 35.1T + 529T^{2} \) |
| 29 | \( 1 + 24.4T + 841T^{2} \) |
| 31 | \( 1 - 37.4iT - 961T^{2} \) |
| 37 | \( 1 - 25.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 3.71iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 3.37iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 40.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 49.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 0.883iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 65.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 86.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 61.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 27.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 65.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 42.2iT - 9.40e3T^{2} \) |
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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
−10.22286927105519201726330927660, −9.614675447578294066171112303196, −8.968908521956468094882481386205, −8.023171360388413597295089213614, −7.56726995265413531978701517420, −6.26872257349318765204022446615, −5.34933484176220570502172687092, −3.88650901796273289683892113962, −2.52993938198524135802133659732, −1.15520348716861274274214709963,
0.18900218288418468605360274026, 1.67234613157416616217115100555, 2.48838476933225977109139078722, 4.09954209290173921331651602931, 6.02920446408254343573532730391, 6.58906389549101344897866193099, 7.51944226321493828242058309290, 8.087969198674185336157838681873, 9.094745526452580746757499609726, 9.593662686464598622830645908626