| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 5·11-s − 2·13-s − 14-s + 16-s − 2·19-s − 20-s − 5·22-s + 4·23-s − 4·25-s + 2·26-s + 28-s − 9·29-s + 9·31-s − 32-s − 35-s + 10·37-s + 2·38-s + 40-s + 8·41-s − 10·43-s + 5·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.50·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 1.06·22-s + 0.834·23-s − 4/5·25-s + 0.392·26-s + 0.188·28-s − 1.67·29-s + 1.61·31-s − 0.176·32-s − 0.169·35-s + 1.64·37-s + 0.324·38-s + 0.158·40-s + 1.24·41-s − 1.52·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.598224384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598224384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.94818428517793, −14.55923690738573, −13.98120979365388, −13.30161452449052, −12.68480476074087, −12.08682840589491, −11.65767133437237, −11.20356059559988, −10.82982648775240, −9.886324808826630, −9.580204163391498, −9.069866156981306, −8.451553466521331, −7.938261501595585, −7.340864533609588, −6.922395591476162, −6.150920450904977, −5.777909142776407, −4.729015677469329, −4.265663948450100, −3.645227042866910, −2.812702439103863, −2.066646328587330, −1.313289885368500, −0.5661775369845946,
0.5661775369845946, 1.313289885368500, 2.066646328587330, 2.812702439103863, 3.645227042866910, 4.265663948450100, 4.729015677469329, 5.777909142776407, 6.150920450904977, 6.922395591476162, 7.340864533609588, 7.938261501595585, 8.451553466521331, 9.069866156981306, 9.580204163391498, 9.886324808826630, 10.82982648775240, 11.20356059559988, 11.65767133437237, 12.08682840589491, 12.68480476074087, 13.30161452449052, 13.98120979365388, 14.55923690738573, 14.94818428517793
