L(s) = 1 | + i·2-s + 2·3-s + 4-s + 2i·6-s + 3i·8-s + 9-s − 2i·11-s + 2·12-s + (−2 + 3i)13-s − 16-s + i·18-s − 6i·19-s + 2·22-s + 6·23-s + 6i·24-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.15·3-s + 0.5·4-s + 0.816i·6-s + 1.06i·8-s + 0.333·9-s − 0.603i·11-s + 0.577·12-s + (−0.554 + 0.832i)13-s − 0.250·16-s + 0.235i·18-s − 1.37i·19-s + 0.426·22-s + 1.25·23-s + 1.22i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82580 + 0.977140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82580 + 0.977140i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 8iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−11.51097974961019453693761398593, −11.04126496168835771267326474181, −9.432934503057887530189474089897, −8.886620010724159412422674149030, −7.83657285223277931890250951501, −7.14920297447368846528732085139, −6.09016604512008503144563533649, −4.79770767370664352185513698856, −3.20114426675046430668429134682, −2.19968394585130162988212608852,
1.77144920368671874843295631541, 2.90609290108933831304524676338, 3.73961609501620932225141553181, 5.39606012788722682752404915554, 6.91087513343003390718548320993, 7.72138239672211531780943250613, 8.723785406429474636325079626613, 9.793977656800040321730176468921, 10.38546052576691879232312697168, 11.48335480489018205382092358843