L(s) = 1 | + 2-s − 2.80·3-s + 4-s + 5-s − 2.80·6-s + 3.22·7-s + 8-s + 4.84·9-s + 10-s + 0.229·11-s − 2.80·12-s + 13-s + 3.22·14-s − 2.80·15-s + 16-s − 6.71·17-s + 4.84·18-s − 4.55·19-s + 20-s − 9.02·21-s + 0.229·22-s − 6.54·23-s − 2.80·24-s + 25-s + 26-s − 5.16·27-s + 3.22·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.447·5-s − 1.14·6-s + 1.21·7-s + 0.353·8-s + 1.61·9-s + 0.316·10-s + 0.0691·11-s − 0.808·12-s + 0.277·13-s + 0.861·14-s − 0.723·15-s + 0.250·16-s − 1.62·17-s + 1.14·18-s − 1.04·19-s + 0.223·20-s − 1.96·21-s + 0.0488·22-s − 1.36·23-s − 0.571·24-s + 0.200·25-s + 0.196·26-s − 0.993·27-s + 0.608·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 - 0.229T + 11T^{2} \) |
| 17 | \( 1 + 6.71T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 + 6.54T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 37 | \( 1 + 0.515T + 37T^{2} \) |
| 41 | \( 1 + 3.14T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.04T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 - 5.96T + 67T^{2} \) |
| 71 | \( 1 - 6.27T + 71T^{2} \) |
| 73 | \( 1 - 5.07T + 73T^{2} \) |
| 79 | \( 1 + 4.86T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 5.22T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.989906166612902628884382317275, −6.91374960813184615604664851767, −6.39317153833760522882257232842, −5.85531554710376999064101360220, −4.96588941882004924479894413051, −4.64338491271236818082708040051, −3.81383936454106582223647305312, −2.16636090200251347684431665594, −1.57591016972136625545720678030, 0,
1.57591016972136625545720678030, 2.16636090200251347684431665594, 3.81383936454106582223647305312, 4.64338491271236818082708040051, 4.96588941882004924479894413051, 5.85531554710376999064101360220, 6.39317153833760522882257232842, 6.91374960813184615604664851767, 7.989906166612902628884382317275