L(s) = 1 | + (0.448 − 0.258i)5-s + (0.866 + 1.5i)13-s + 1.93i·17-s + (−0.366 + 0.633i)25-s + (−1.67 − 0.965i)29-s + 37-s + (1.22 − 0.707i)41-s + (0.5 + 0.866i)49-s − 1.41i·53-s + (0.5 − 0.866i)61-s + (0.776 + 0.448i)65-s + 1.73·73-s + (0.499 + 0.866i)85-s + 0.517i·89-s + (−1.22 − 0.707i)101-s + ⋯ |
L(s) = 1 | + (0.448 − 0.258i)5-s + (0.866 + 1.5i)13-s + 1.93i·17-s + (−0.366 + 0.633i)25-s + (−1.67 − 0.965i)29-s + 37-s + (1.22 − 0.707i)41-s + (0.5 + 0.866i)49-s − 1.41i·53-s + (0.5 − 0.866i)61-s + (0.776 + 0.448i)65-s + 1.73·73-s + (0.499 + 0.866i)85-s + 0.517i·89-s + (−1.22 − 0.707i)101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.323275672\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323275672\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.93iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.73T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - 0.517iT - T^{2} \) |
| 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
−9.330984800832234277718495984077, −8.387700045909132893324434185972, −7.74595891764707767042332828669, −6.67734404244456466002034529203, −6.06726830731828025646944454993, −5.41995431055724860221592878372, −4.07653774983555271886413719405, −3.84211248126878035571129925807, −2.20240455667566436624183582313, −1.50508481208197083349854911807,
0.948367824448422089836272258093, 2.43771726693507515393862933515, 3.14410155713432365767001056232, 4.17717069620946883856514485815, 5.32655239942626429691933872333, 5.74709627100314517748282562868, 6.70733588585617104384980264450, 7.53098470584159558638377819745, 8.123666949418544241499554436077, 9.173535575881785897786903660623