L(s) = 1 | + 1.73i·5-s + 1.73·7-s − 9-s + i·11-s − 17-s − i·19-s − 1.99·25-s + 2.99i·35-s − i·43-s − 1.73i·45-s + 1.73·47-s + 1.99·49-s − 1.73·55-s + 1.73i·61-s − 1.73·63-s + ⋯ |
L(s) = 1 | + 1.73i·5-s + 1.73·7-s − 9-s + i·11-s − 17-s − i·19-s − 1.99·25-s + 2.99i·35-s − i·43-s − 1.73i·45-s + 1.73·47-s + 1.99·49-s − 1.73·55-s + 1.73i·61-s − 1.73·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.125899593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.125899593\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 - 1.73iT - T^{2} \) |
| 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 - 1.73T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 1.73iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 2iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.48698786022753658915608073535, −9.204714783875373542955193355639, −8.429447277324166384827487973601, −7.44241430682648214293199617657, −7.02422408917260611142478651111, −5.96936623512845364835318545403, −4.97994506259223041681830704473, −4.05939879201996381022235652347, −2.68253133036454636010506112612, −2.08601991965922016682182788152,
1.06166890326627544015399119234, 2.18897370204960924819179293151, 3.86138599817062274226591239329, 4.77758906319748045592440684855, 5.37173359460988602407981346933, 6.10944500377023084003472926039, 7.73874528907677206285809350335, 8.365483112523258028602134831817, 8.644790175219062476372765591033, 9.468393245248311978610028697593