| L(s) = 1 | − 1.74·2-s + 0.824·3-s + 1.05·4-s − 1.38·5-s − 1.44·6-s + 2.70·7-s + 1.65·8-s − 2.31·9-s + 2.42·10-s − 11-s + 0.869·12-s − 4.73·14-s − 1.14·15-s − 4.99·16-s + 6.27·17-s + 4.05·18-s − 5.86·19-s − 1.46·20-s + 2.23·21-s + 1.74·22-s − 3.21·23-s + 1.36·24-s − 3.07·25-s − 4.38·27-s + 2.85·28-s + 2.53·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.476·3-s + 0.526·4-s − 0.620·5-s − 0.588·6-s + 1.02·7-s + 0.584·8-s − 0.773·9-s + 0.766·10-s − 0.301·11-s + 0.250·12-s − 1.26·14-s − 0.295·15-s − 1.24·16-s + 1.52·17-s + 0.955·18-s − 1.34·19-s − 0.326·20-s + 0.487·21-s + 0.372·22-s − 0.669·23-s + 0.278·24-s − 0.614·25-s − 0.844·27-s + 0.539·28-s + 0.470·29-s + 0.365·30-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 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 - 0.824T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 17 | \( 1 - 6.27T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 8.95T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 + 8.74T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 + 2.15T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.492709652736321053807243822566, −8.205176484521623443775112209293, −7.85564074524551390795347385618, −6.82830499267195184148072480680, −5.63688763857098562136013659720, −4.68158123419165898854318179883, −3.75021814745422503667749609494, −2.51057805756129295628014911480, −1.44638639641366593149325890766, 0,
1.44638639641366593149325890766, 2.51057805756129295628014911480, 3.75021814745422503667749609494, 4.68158123419165898854318179883, 5.63688763857098562136013659720, 6.82830499267195184148072480680, 7.85564074524551390795347385618, 8.205176484521623443775112209293, 8.492709652736321053807243822566
