| L(s) = 1 | − 2i·7-s − 2i·13-s + 6i·17-s − 4·19-s − 6i·23-s + 6·29-s + 4·31-s + 2i·37-s − 6·41-s − 10i·43-s − 6i·47-s + 3·49-s − 6i·53-s − 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s − 0.554i·13-s + 1.45i·17-s − 0.917·19-s − 1.25i·23-s + 1.11·29-s + 0.718·31-s + 0.328i·37-s − 0.937·41-s − 1.52i·43-s − 0.875i·47-s + 0.428·49-s − 0.824i·53-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174373858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174373858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.414165381697922182188769224247, −7.65729836594056680966072047181, −6.65206196881078302073826784167, −6.31132885532463616513311594652, −5.23223852850501315038951118413, −4.37530415596092419385072835092, −3.74435783615640653173457378257, −2.70211960657649289090758403259, −1.61575738787133610530147912047, −0.35611247292156653907058253133,
1.28209248960265277118192318987, 2.45440663784945304908058781459, 3.11805689524617042229945770576, 4.34343037998614597722744289521, 4.94029586276568208421997033709, 5.85764403975763330722991810024, 6.51995456507046611306335274753, 7.31984579911223066795033079662, 8.079069396905827445691607462935, 8.884850748669084902261967434104
