| L(s) = 1 | − i·2-s + i·3-s − 4-s + (−0.224 − 2.22i)5-s + 6-s − 3.44i·7-s + i·8-s − 9-s + (−2.22 + 0.224i)10-s + 1.44·11-s − i·12-s + 0.550i·13-s − 3.44·14-s + (2.22 − 0.224i)15-s + 16-s − 3.44i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.100 − 0.994i)5-s + 0.408·6-s − 1.30i·7-s + 0.353i·8-s − 0.333·9-s + (−0.703 + 0.0710i)10-s + 0.437·11-s − 0.288i·12-s + 0.152i·13-s − 0.921·14-s + (0.574 − 0.0580i)15-s + 0.250·16-s − 0.836i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9433558566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9433558566\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (0.224 + 2.22i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.44iT - 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 - 0.550iT - 13T^{2} \) |
| 17 | \( 1 + 3.44iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 41 | \( 1 + 0.449T + 41T^{2} \) |
| 43 | \( 1 + 6.89iT - 43T^{2} \) |
| 47 | \( 1 - 6.44iT - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 6.89iT - 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 + 8.79iT - 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 1.44iT - 83T^{2} \) |
| 89 | \( 1 + 0.348T + 89T^{2} \) |
| 97 | \( 1 + 3.34iT - 97T^{2} \) |
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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
−9.458343046925194584812956105411, −8.951488296000469809500745999614, −7.958711954326663930865211706019, −7.08458014835453609876088330698, −5.78863081029412090435757443259, −4.70851676198374917926048635415, −4.20853763565165879947857924853, −3.32833039938196138978049009932, −1.70328877473667172111999358134, −0.41200896554588044472365426993,
1.91030552513461033261332546994, 2.98402082858061395725993369768, 4.14262289777883781673982093007, 5.55041574340890509808309767954, 6.12425294997650808160273060812, 6.80895183879384306910343152432, 7.70064808938716857367545159136, 8.489044349460640158191375467408, 9.155745953970287897492751160703, 10.17134546007882411844190022081
