| L(s) = 1 | − 1.85·2-s − 2.66·3-s + 1.44·4-s + 5-s + 4.94·6-s + 7-s + 1.03·8-s + 4.11·9-s − 1.85·10-s − 3.84·12-s + 1.71·13-s − 1.85·14-s − 2.66·15-s − 4.80·16-s + 4.34·17-s − 7.63·18-s + 3.07·19-s + 1.44·20-s − 2.66·21-s + 1.16·23-s − 2.76·24-s + 25-s − 3.18·26-s − 2.97·27-s + 1.44·28-s − 10.0·29-s + 4.94·30-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 1.54·3-s + 0.720·4-s + 0.447·5-s + 2.02·6-s + 0.377·7-s + 0.366·8-s + 1.37·9-s − 0.586·10-s − 1.10·12-s + 0.476·13-s − 0.495·14-s − 0.688·15-s − 1.20·16-s + 1.05·17-s − 1.79·18-s + 0.705·19-s + 0.322·20-s − 0.582·21-s + 0.242·23-s − 0.564·24-s + 0.200·25-s − 0.625·26-s − 0.572·27-s + 0.272·28-s − 1.86·29-s + 0.903·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 3 | \( 1 + 2.66T + 3T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 + 8.91T + 37T^{2} \) |
| 41 | \( 1 + 2.96T + 41T^{2} \) |
| 43 | \( 1 + 6.91T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 - 2.96T + 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 9.84T + 61T^{2} \) |
| 67 | \( 1 - 0.429T + 67T^{2} \) |
| 71 | \( 1 + 2.04T + 71T^{2} \) |
| 73 | \( 1 + 3.45T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 1.17T + 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.029736438211299848202885283814, −7.33079584337915709358876678041, −6.69970309912246339269482299082, −5.81775741051689085891832627204, −5.28433919257811444371528031445, −4.54280342750461345737959531011, −3.32169743780096132031456777092, −1.74065980740364992234321366185, −1.14687803969888196417407469471, 0,
1.14687803969888196417407469471, 1.74065980740364992234321366185, 3.32169743780096132031456777092, 4.54280342750461345737959531011, 5.28433919257811444371528031445, 5.81775741051689085891832627204, 6.69970309912246339269482299082, 7.33079584337915709358876678041, 8.029736438211299848202885283814
