L(s) = 1 | + 1.41·2-s + (−2.70 + 1.28i)3-s + 2.00·4-s + (3.71 − 3.34i)5-s + (−3.83 + 1.82i)6-s − 2.64i·7-s + 2.82·8-s + (5.68 − 6.97i)9-s + (5.25 − 4.73i)10-s − 10.4i·11-s + (−5.41 + 2.57i)12-s + 11.9i·13-s − 3.74i·14-s + (−5.76 + 13.8i)15-s + 4.00·16-s + 29.1·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.903 + 0.429i)3-s + 0.500·4-s + (0.743 − 0.669i)5-s + (−0.638 + 0.303i)6-s − 0.377i·7-s + 0.353·8-s + (0.631 − 0.775i)9-s + (0.525 − 0.473i)10-s − 0.951i·11-s + (−0.451 + 0.214i)12-s + 0.922i·13-s − 0.267i·14-s + (−0.384 + 0.923i)15-s + 0.250·16-s + 1.71·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.00794 - 0.401107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00794 - 0.401107i\) |
\(L(2)\) |
|
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 | \( 1 + (2.70 - 1.28i)T \) |
| 5 | \( 1 + (-3.71 + 3.34i)T \) |
| 7 | \( 1 + 2.64iT \) |
good | 11 | \( 1 + 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 11.9iT - 169T^{2} \) |
| 17 | \( 1 - 29.1T + 289T^{2} \) |
| 19 | \( 1 - 20.3T + 361T^{2} \) |
| 23 | \( 1 + 7.71T + 529T^{2} \) |
| 29 | \( 1 + 47.4iT - 841T^{2} \) |
| 31 | \( 1 + 35.5T + 961T^{2} \) |
| 37 | \( 1 - 58.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 52.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 85.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 42.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 37.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 58.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 19.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 33.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 46.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 65.1iT - 9.40e3T^{2} \) |
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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
−11.93477341825700836434902648065, −11.43583910193675579751097294154, −10.11444953755672876516138350572, −9.519324128800628461052095365194, −7.939525622661138367904297364127, −6.46461585454728961690137751918, −5.67004768739232519947067877226, −4.78219611440005113824128634886, −3.49672971646216745538348362949, −1.22595378934804772589826004231,
1.71782247084584355952383265516, 3.28664089651936463820617266545, 5.27588409465921985734301878846, 5.64170613132741133077712999881, 6.95845469279271946358856126216, 7.66543815713444913398438930616, 9.662858927668091061927500086670, 10.42899329974656467259407464216, 11.35340186217906469451826849117, 12.50187519229633028029392980665