L(s) = 1 | + 1.22·2-s − 2.50·3-s − 0.507·4-s + 2.28·5-s − 3.06·6-s + 0.285·7-s − 3.06·8-s + 3.28·9-s + 2.79·10-s − 11-s + 1.27·12-s + 0.348·14-s − 5.72·15-s − 2.72·16-s + 0.507·17-s + 4.01·18-s + 3.95·19-s − 1.15·20-s − 0.714·21-s − 1.22·22-s − 0.0632·23-s + 7.67·24-s + 0.221·25-s − 0.714·27-s − 0.144·28-s − 10.7·29-s − 6.99·30-s + ⋯ |
L(s) = 1 | + 0.863·2-s − 1.44·3-s − 0.253·4-s + 1.02·5-s − 1.25·6-s + 0.107·7-s − 1.08·8-s + 1.09·9-s + 0.882·10-s − 0.301·11-s + 0.366·12-s + 0.0931·14-s − 1.47·15-s − 0.682·16-s + 0.122·17-s + 0.946·18-s + 0.906·19-s − 0.259·20-s − 0.155·21-s − 0.260·22-s − 0.0131·23-s + 1.56·24-s + 0.0443·25-s − 0.137·27-s − 0.0273·28-s − 1.99·29-s − 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 - 0.285T + 7T^{2} \) |
| 17 | \( 1 - 0.507T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + 0.0632T + 23T^{2} \) |
| 29 | \( 1 + 10.7T + 29T^{2} \) |
| 31 | \( 1 - 3.57T + 31T^{2} \) |
| 37 | \( 1 + 7.72T + 37T^{2} \) |
| 41 | \( 1 - 0.380T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 + 0.207T + 67T^{2} \) |
| 71 | \( 1 - 2.77T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 3.45T + 79T^{2} \) |
| 83 | \( 1 + 4.74T + 83T^{2} \) |
| 89 | \( 1 - 3.71T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.161293278010716322046388847952, −7.918021529241501660646281982119, −6.83051911991716502682174580394, −6.07633850420625022011302226494, −5.44285261731920590721213538170, −5.14915686582588892105789225291, −4.12000906399521823034943651220, −2.99297796205471142203649578613, −1.56425551608615543001246249097, 0,
1.56425551608615543001246249097, 2.99297796205471142203649578613, 4.12000906399521823034943651220, 5.14915686582588892105789225291, 5.44285261731920590721213538170, 6.07633850420625022011302226494, 6.83051911991716502682174580394, 7.918021529241501660646281982119, 9.161293278010716322046388847952