L(s) = 1 | + 2-s + 1.74·3-s + 4-s − 2.50·5-s + 1.74·6-s + 1.18·7-s + 8-s + 0.0572·9-s − 2.50·10-s + 1.65·11-s + 1.74·12-s + 1.18·14-s − 4.38·15-s + 16-s − 5.69·17-s + 0.0572·18-s − 19-s − 2.50·20-s + 2.07·21-s + 1.65·22-s − 1.77·23-s + 1.74·24-s + 1.28·25-s − 5.14·27-s + 1.18·28-s − 2.45·29-s − 4.38·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.00·3-s + 0.5·4-s − 1.12·5-s + 0.713·6-s + 0.448·7-s + 0.353·8-s + 0.0190·9-s − 0.792·10-s + 0.499·11-s + 0.504·12-s + 0.316·14-s − 1.13·15-s + 0.250·16-s − 1.38·17-s + 0.0134·18-s − 0.229·19-s − 0.560·20-s + 0.452·21-s + 0.353·22-s − 0.370·23-s + 0.356·24-s + 0.257·25-s − 0.990·27-s + 0.224·28-s − 0.456·29-s − 0.800·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 2.45T + 29T^{2} \) |
| 31 | \( 1 + 3.34T + 31T^{2} \) |
| 37 | \( 1 + 4.12T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.86T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 0.571T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.11T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 + 0.161T + 97T^{2} \) |
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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
−7.68872796439948826386698250247, −7.06676138311469902604284197116, −6.35291689003241598143263094118, −5.36971119408545109914369673996, −4.54317539954884545955855511210, −3.87859206100229848554491762247, −3.44569832215861055082238385346, −2.43704708241167906827137536429, −1.71390073982432671155835352443, 0,
1.71390073982432671155835352443, 2.43704708241167906827137536429, 3.44569832215861055082238385346, 3.87859206100229848554491762247, 4.54317539954884545955855511210, 5.36971119408545109914369673996, 6.35291689003241598143263094118, 7.06676138311469902604284197116, 7.68872796439948826386698250247