L(s) = 1 | + (−0.385 + 0.222i)2-s + (−0.400 + 0.694i)4-s + (0.866 + 0.5i)7-s − 0.801i·8-s + (−0.5 + 0.866i)9-s + (1.56 − 0.900i)11-s − 0.445·14-s + (−0.222 − 0.385i)16-s − 0.445i·18-s + (−0.400 + 0.694i)22-s + (0.623 + 1.07i)23-s − 25-s + (−0.694 + 0.400i)28-s + (0.222 + 0.385i)29-s + (0.866 + 0.5i)32-s + ⋯ |
L(s) = 1 | + (−0.385 + 0.222i)2-s + (−0.400 + 0.694i)4-s + (0.866 + 0.5i)7-s − 0.801i·8-s + (−0.5 + 0.866i)9-s + (1.56 − 0.900i)11-s − 0.445·14-s + (−0.222 − 0.385i)16-s − 0.445i·18-s + (−0.400 + 0.694i)22-s + (0.623 + 1.07i)23-s − 25-s + (−0.694 + 0.400i)28-s + (0.222 + 0.385i)29-s + (0.866 + 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8716284717\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8716284717\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (1.07 + 0.623i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.922751642636101834626434388820, −8.945664410501638777934891779113, −8.617435103760435895268647320903, −7.84913170028380383177473570350, −7.02524153125089874645721472931, −5.92477894965148526989001043688, −5.03815135413188061523627056849, −4.00080837025983562053803127634, −3.06302537342036881338581088784, −1.55901737734784306386087592257,
1.04004039987158459117446968476, 2.11451130728439323248104217928, 3.84750934026940799805983330009, 4.52184198802609399034289133229, 5.56466303761406788182584445225, 6.50908490216435013740805883677, 7.28658566182619898027126271843, 8.567937032519286405369491202864, 8.926321192008666423620522157463, 9.834727070146322422211867172272