| L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 3·9-s − 11-s − 6i·13-s + 14-s + 16-s + 2i·17-s + 3i·18-s + 4·19-s − i·22-s − 4i·23-s + 6·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s + 9-s − 0.301·11-s − 1.66i·13-s + 0.267·14-s + 0.250·16-s + 0.485i·17-s + 0.707i·18-s + 0.917·19-s − 0.213i·22-s − 0.834i·23-s + 1.17·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.333353410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333353410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 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
−8.116168086031270133034072604417, −7.60320121248209902541660160773, −7.03493154513350502196768180668, −6.14074484078343871786561989133, −5.39493589315366128708080559518, −4.72826074384213555610858009550, −3.79346674154845856293098479986, −3.03108319879693138833611298753, −1.59151511905736036998980673442, −0.40202244269629993532774533201,
1.33769913024297275647513031781, 2.02067499039103804188448276726, 3.11960013778609595436655279504, 3.98129900574970572151715817350, 4.70543224531536784624837250795, 5.44179886515074608210095222841, 6.44186442413218420803964284218, 7.26439886544233776522273756892, 7.80879582271886806233556315332, 8.927491307688733560001325783109
