| L(s) = 1 | − i·2-s − 4-s − 3.46·5-s + i·8-s + 3.46i·10-s − 5.19i·13-s + 16-s + 6.92·17-s − 3.46i·19-s + 3.46·20-s + 6i·23-s + 6.99·25-s − 5.19·26-s − 8.66i·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.54·5-s + 0.353i·8-s + 1.09i·10-s − 1.44i·13-s + 0.250·16-s + 1.68·17-s − 0.794i·19-s + 0.774·20-s + 1.25i·23-s + 1.39·25-s − 1.01·26-s − 1.55i·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3292703542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3292703542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 1.73iT - 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 1.73iT - 97T^{2} \) |
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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
−8.265976276907389672256042942026, −7.72175791494154791931235313773, −7.27970964689231312660900072500, −5.81920775758457394669622206698, −5.16312677064791040863063643269, −4.14789006328909177337762873711, −3.42590930339096614781817055359, −2.85459611897854463517318933179, −1.19528883394812666389993654710, −0.12958975186300106178134200137,
1.39723544372457057026105391958, 3.13316084887082787238013189480, 3.88345623291407060080117300649, 4.56135623003527393968994455505, 5.41738614774005777433898050750, 6.49285415298542086026368191696, 7.11963018828091905526863657593, 7.75323975324952347785911926557, 8.462859035070152054917353384278, 8.956247881320531108922369554054
