L(s) = 1 | − 1.41i·2-s + 1.73·3-s − 2.00·4-s − 2.44i·6-s + (4.78 + 5.11i)7-s + 2.82i·8-s + 2.99·9-s + 10.3·11-s − 3.46·12-s − 0.605·13-s + (7.23 − 6.76i)14-s + 4.00·16-s − 5.49·17-s − 4.24i·18-s + 33.2i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.500·4-s − 0.408i·6-s + (0.683 + 0.730i)7-s + 0.353i·8-s + 0.333·9-s + 0.941·11-s − 0.288·12-s − 0.0465·13-s + (0.516 − 0.483i)14-s + 0.250·16-s − 0.323·17-s − 0.235i·18-s + 1.74i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.369613658\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.369613658\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-4.78 - 5.11i)T \) |
good | 11 | \( 1 - 10.3T + 121T^{2} \) |
| 13 | \( 1 + 0.605T + 169T^{2} \) |
| 17 | \( 1 + 5.49T + 289T^{2} \) |
| 19 | \( 1 - 33.2iT - 361T^{2} \) |
| 23 | \( 1 - 14.4iT - 529T^{2} \) |
| 29 | \( 1 + 42.0T + 841T^{2} \) |
| 31 | \( 1 - 0.540iT - 961T^{2} \) |
| 37 | \( 1 + 13.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 19.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 29.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 12.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 42.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 66.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 27.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 126.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 30.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 164.T + 9.40e3T^{2} \) |
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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
−9.515517740702534282986391203487, −9.181871683231082094902037893261, −8.200346138039689438333877693492, −7.59230006317561502584733488980, −6.23370478684620860953775212826, −5.35421401753215531464259686569, −4.19850972535440953951396527802, −3.46301135759824425872379631058, −2.17810440125438323215513491151, −1.40870505022969172174001125839,
0.72791751325180234360805360969, 2.16345904694046347778903639252, 3.65902351477540198786305191479, 4.41370758507627935232533281188, 5.29392514595251090770948496300, 6.62748687839199517160827699819, 7.12863924294289979027330486992, 7.958465563368021793602420445145, 8.891078326480976879111158167409, 9.296042230329369815166956708347