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
−9.468393245248311978610028697593, −8.644790175219062476372765591033, −8.365483112523258028602134831817, −7.73874528907677206285809350335, −6.10944500377023084003472926039, −5.37173359460988602407981346933, −4.77758906319748045592440684855, −3.86138599817062274226591239329, −2.18897370204960924819179293151, −1.06166890326627544015399119234,
2.08601991965922016682182788152, 2.68253133036454636010506112612, 4.05939879201996381022235652347, 4.97994506259223041681830704473, 5.96936623512845364835318545403, 7.02422408917260611142478651111, 7.44241430682648214293199617657, 8.429447277324166384827487973601, 9.204714783875373542955193355639, 10.48698786022753658915608073535