L(s) = 1 | + (1 − i)2-s − 2i·4-s − 3i·5-s + 3·7-s + (−2 − 2i)8-s + (−3 − 3i)10-s + i·13-s + (3 − 3i)14-s − 4·16-s + 7·17-s − 4i·19-s − 6·20-s − 4·23-s − 4·25-s + (1 + i)26-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s − 1.34i·5-s + 1.13·7-s + (−0.707 − 0.707i)8-s + (−0.948 − 0.948i)10-s + 0.277i·13-s + (0.801 − 0.801i)14-s − 16-s + 1.69·17-s − 0.917i·19-s − 1.34·20-s − 0.834·23-s − 0.800·25-s + (0.196 + 0.196i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.953463 - 2.30186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.953463 - 2.30186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.810929708715935350486376354643, −9.002976634679524268556987485372, −8.256892312600616901149994718986, −7.23009806150085706545676071888, −5.80182269904285993340750573937, −5.12351612177070882626560888552, −4.54168481132733101164581341578, −3.45500436521338596356623048772, −1.91542847886084698137045959320, −1.01251428109686507359472261915,
2.07875543361596919789728301552, 3.30337636812120140812732157124, 4.06336662775372832913137138087, 5.41375683031812336635310272242, 5.90549497669378764643639949139, 7.04750305861168250111128132207, 7.74795126446030632420585194532, 8.189352302905368400678882268362, 9.568344033548428929447615449757, 10.54879325857465538070230369061