| L(s) = 1 | + 2.44·2-s − 2.44·3-s + 3.99·4-s − 2.44·5-s − 5.99·6-s + 2.44·7-s + 4.89·8-s + 2.99·9-s − 5.99·10-s − 11-s − 9.79·12-s − 3·13-s + 5.99·14-s + 5.99·15-s + 3.99·16-s − 7·17-s + 7.34·18-s + 4.89·19-s − 9.79·20-s − 5.99·21-s − 2.44·22-s + 23-s − 11.9·24-s + 0.999·25-s − 7.34·26-s + 9.79·28-s − 2.44·29-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 1.41·3-s + 1.99·4-s − 1.09·5-s − 2.44·6-s + 0.925·7-s + 1.73·8-s + 0.999·9-s − 1.89·10-s − 0.301·11-s − 2.82·12-s − 0.832·13-s + 1.60·14-s + 1.54·15-s + 0.999·16-s − 1.69·17-s + 1.73·18-s + 1.12·19-s − 2.19·20-s − 1.30·21-s − 0.522·22-s + 0.208·23-s − 2.44·24-s + 0.199·25-s − 1.44·26-s + 1.85·28-s − 0.454·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 7.34T + 61T^{2} \) |
| 67 | \( 1 - 9T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
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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
−8.649842777033441119852664742335, −7.53559212230323752887327208127, −7.04407308951021410917296057822, −6.21377957399014347954774776352, −5.19349748322025490914723346512, −4.91963650750806191459270266519, −4.23292310430270266036539560339, −3.21098522528394051622634132865, −1.88726173175353408736274236765, 0,
1.88726173175353408736274236765, 3.21098522528394051622634132865, 4.23292310430270266036539560339, 4.91963650750806191459270266519, 5.19349748322025490914723346512, 6.21377957399014347954774776352, 7.04407308951021410917296057822, 7.53559212230323752887327208127, 8.649842777033441119852664742335
