L(s) = 1 | − 2-s − 1.41i·3-s + 4-s + 1.41i·6-s + 2.64·7-s − 8-s + 0.999·9-s + 3.74i·11-s − 1.41i·12-s − 4.24i·13-s − 2.64·14-s + 16-s − 0.999·18-s + 5.29·19-s − 3.74i·21-s − 3.74i·22-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.816i·3-s + 0.5·4-s + 0.577i·6-s + 0.999·7-s − 0.353·8-s + 0.333·9-s + 1.12i·11-s − 0.408i·12-s − 1.17i·13-s − 0.707·14-s + 0.250·16-s − 0.235·18-s + 1.21·19-s − 0.816i·21-s − 0.797i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.993511 - 0.476841i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.993511 - 0.476841i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - 2.64T \) |
| 23 | \( 1 + (3 + 3.74i)T \) |
good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 3.74iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 3.74iT - 53T^{2} \) |
| 59 | \( 1 - 9.89iT - 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
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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
−11.70266818889211399252020611156, −10.31413132067062923029155585296, −9.829932745128121299017770580692, −8.327295884818461448551047948773, −7.75599911046148835035416523604, −7.02029591069777022514908896740, −5.74522891009708330016810871975, −4.42682126234738428855271684586, −2.46226424312838764121391980359, −1.21919702188197437885948785301,
1.58868820700475365762621695545, 3.46409753294503503203626545132, 4.66772228430256113542159063911, 5.81447876427472224402836804844, 7.17715271022436983984392441292, 8.122299399584182675439812938478, 9.095867216632234584035443687028, 9.778110192766899702997657322637, 10.86085359902902181695477033907, 11.38576555117347512121276627570