L(s) = 1 | − 2-s + 4-s + (2.23 + 1.41i)7-s − 8-s + 1.41i·11-s − 0.926·13-s + (−2.23 − 1.41i)14-s + 16-s − 2.23i·17-s + 7.63i·19-s − 1.41i·22-s − 23-s + 0.926·26-s + (2.23 + 1.41i)28-s − 0.757i·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.845 + 0.534i)7-s − 0.353·8-s + 0.426i·11-s − 0.256·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s − 0.542i·17-s + 1.75i·19-s − 0.301i·22-s − 0.208·23-s + 0.181·26-s + (0.422 + 0.267i)28-s − 0.140i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9483898453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9483898453\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.23 - 1.41i)T \) |
good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 0.926T + 13T^{2} \) |
| 17 | \( 1 + 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 7.63iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 0.757iT - 29T^{2} \) |
| 31 | \( 1 - 4.08iT - 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 + 8.56T + 41T^{2} \) |
| 43 | \( 1 - 3.58iT - 43T^{2} \) |
| 47 | \( 1 - 1.30iT - 47T^{2} \) |
| 53 | \( 1 + 8.07T + 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 - 0.926iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + 2.61T + 89T^{2} \) |
| 97 | \( 1 - 0.542T + 97T^{2} \) |
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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
−8.819238795012160594496714381271, −8.211112509502681502036894091711, −7.62971377458995903848049873946, −6.85529702980237060723918229342, −5.90942847690246059860358145326, −5.22317643381144514232734228281, −4.31291440431150078416415210713, −3.19743576660060655518440465008, −2.12739128706677589010360492127, −1.37111067517537875049375871336,
0.37439406263420404213465104189, 1.53998293753685243949603595295, 2.53881126697437707771702740197, 3.61261201988250633661110299621, 4.62488176316346280855987206900, 5.33623213590807010676580988584, 6.41472676102494962913268374784, 7.04438619575927199140529529398, 7.81623099936799131156541678778, 8.418804261021071607415583862915