L(s) = 1 | − 0.552·2-s − 3-s − 1.69·4-s − 1.59·5-s + 0.552·6-s + 2.04·8-s + 9-s + 0.878·10-s − 11-s + 1.69·12-s − 2.87·13-s + 1.59·15-s + 2.26·16-s − 4.83·17-s − 0.552·18-s + 1.14·19-s + 2.69·20-s + 0.552·22-s − 3.65·23-s − 2.04·24-s − 2.47·25-s + 1.59·26-s − 27-s − 0.325·29-s − 0.878·30-s − 6.45·31-s − 5.33·32-s + ⋯ |
L(s) = 1 | − 0.390·2-s − 0.577·3-s − 0.847·4-s − 0.711·5-s + 0.225·6-s + 0.721·8-s + 0.333·9-s + 0.277·10-s − 0.301·11-s + 0.489·12-s − 0.798·13-s + 0.410·15-s + 0.565·16-s − 1.17·17-s − 0.130·18-s + 0.262·19-s + 0.602·20-s + 0.117·22-s − 0.761·23-s − 0.416·24-s − 0.494·25-s + 0.311·26-s − 0.192·27-s − 0.0605·29-s − 0.160·30-s − 1.15·31-s − 0.942·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4061597978\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4061597978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 0.552T + 2T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.325T + 29T^{2} \) |
| 31 | \( 1 + 6.45T + 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 - 4.05T + 41T^{2} \) |
| 43 | \( 1 - 4.62T + 43T^{2} \) |
| 47 | \( 1 + 0.305T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 1.77T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 + 8.40T + 83T^{2} \) |
| 89 | \( 1 - 5.74T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−9.427569316529826420925907758370, −8.639721717204986415046494635280, −7.74463047582261827146545497942, −7.28613727207526383760163359146, −6.10333610487949633578761473343, −5.16181955077748040061998587935, −4.42540078758503128928314453436, −3.69012740312842942997401141877, −2.11984191104285427172541585450, −0.47318212961588927980077226483,
0.47318212961588927980077226483, 2.11984191104285427172541585450, 3.69012740312842942997401141877, 4.42540078758503128928314453436, 5.16181955077748040061998587935, 6.10333610487949633578761473343, 7.28613727207526383760163359146, 7.74463047582261827146545497942, 8.639721717204986415046494635280, 9.427569316529826420925907758370