L(s) = 1 | − i·2-s − 4-s − 2i·7-s + i·8-s − 4i·13-s − 2·14-s + 16-s + 6i·17-s + 7·19-s − 4·26-s + 2i·28-s + 6·29-s − 10·31-s − i·32-s + 6·34-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.755i·7-s + 0.353i·8-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s + 1.45i·17-s + 1.60·19-s − 0.784·26-s + 0.377i·28-s + 1.11·29-s − 1.79·31-s − 0.176i·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.692424725\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.692424725\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−8.238351007039741974527337009330, −7.57722258358808735598727363065, −6.86909677353774157660711830655, −5.69807111918314342837917991461, −5.28887201959665023928439923614, −4.09099822425226783812750964769, −3.60672087724269147412590314271, −2.70865202978629777466092573965, −1.53438325158638567969320821890, −0.59433246210078581894413624233,
1.04347743841135907048745303464, 2.38779048958888832617022952129, 3.25571159188896679213064215776, 4.34608489938324046090043032395, 5.07810895614880171917144638683, 5.70264484843409895762071121840, 6.50318940822930695702618271278, 7.31226451940302336730862685322, 7.69902753231507802638820379407, 8.805856177567859002070517205193