L(s) = 1 | + 7.53i·5-s − 5.18·7-s − 20.8i·11-s + 5.35·13-s − 8.47i·17-s + 4.35·19-s + 22.0i·23-s − 31.7·25-s + 34.4i·29-s + 35.1·31-s − 39.0i·35-s + 15.3·37-s − 19.8i·41-s − 24.9·43-s − 11.0i·47-s + ⋯ |
L(s) = 1 | + 1.50i·5-s − 0.740·7-s − 1.89i·11-s + 0.412·13-s − 0.498i·17-s + 0.229·19-s + 0.956i·23-s − 1.27·25-s + 1.18i·29-s + 1.13·31-s − 1.11i·35-s + 0.413·37-s − 0.484i·41-s − 0.580·43-s − 0.236i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.216746732\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216746732\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35T \) |
good | 5 | \( 1 - 7.53iT - 25T^{2} \) |
| 7 | \( 1 + 5.18T + 49T^{2} \) |
| 11 | \( 1 + 20.8iT - 121T^{2} \) |
| 13 | \( 1 - 5.35T + 169T^{2} \) |
| 17 | \( 1 + 8.47iT - 289T^{2} \) |
| 23 | \( 1 - 22.0iT - 529T^{2} \) |
| 29 | \( 1 - 34.4iT - 841T^{2} \) |
| 31 | \( 1 - 35.1T + 961T^{2} \) |
| 37 | \( 1 - 15.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 90.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 51.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 47.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.34T + 4.48e3T^{2} \) |
| 71 | \( 1 + 30.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 79.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 152.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 22.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 134. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 32.4T + 9.40e3T^{2} \) |
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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
−8.970162384913472960886636960361, −8.083064006909143362438105834547, −7.31300860559648217163405428270, −6.48375662247234589559454488113, −6.11402207464404238667128322690, −5.22737903604706897813627239747, −3.73734741719118250162322293888, −3.22322736063209001111379046893, −2.66014706117901849821044512260, −1.03651755976219233050917529435,
0.32077641275352334171557257772, 1.46268889338297104861101495776, 2.40768877561106115400982631737, 3.76422369237605388185930739825, 4.58608164123654488263013587633, 5.00499953178179859667752214585, 6.17732561502255376075738797608, 6.73188097187887574463719374541, 7.86032804817429323113403403132, 8.321759023601587959145264350934