L(s) = 1 | + 2.14·2-s − 3-s + 2.61·4-s + 1.12·5-s − 2.14·6-s + 1.32·8-s + 9-s + 2.41·10-s + 5.38·11-s − 2.61·12-s − 5.09·13-s − 1.12·15-s − 2.38·16-s − 4.12·17-s + 2.14·18-s − 2.31·19-s + 2.93·20-s + 11.5·22-s − 7.27·23-s − 1.32·24-s − 3.74·25-s − 10.9·26-s − 27-s + 1.41·29-s − 2.41·30-s − 1.68·31-s − 7.77·32-s + ⋯ |
L(s) = 1 | + 1.51·2-s − 0.577·3-s + 1.30·4-s + 0.501·5-s − 0.877·6-s + 0.469·8-s + 0.333·9-s + 0.762·10-s + 1.62·11-s − 0.755·12-s − 1.41·13-s − 0.289·15-s − 0.595·16-s − 0.999·17-s + 0.506·18-s − 0.531·19-s + 0.657·20-s + 2.46·22-s − 1.51·23-s − 0.271·24-s − 0.748·25-s − 2.14·26-s − 0.192·27-s + 0.262·29-s − 0.440·30-s − 0.302·31-s − 1.37·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 - 1.12T + 5T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 1.68T + 31T^{2} \) |
| 37 | \( 1 + 6.86T + 37T^{2} \) |
| 43 | \( 1 + 1.11T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 + 6.42T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 6.69T + 67T^{2} \) |
| 71 | \( 1 - 2.41T + 71T^{2} \) |
| 73 | \( 1 - 8.85T + 73T^{2} \) |
| 79 | \( 1 - 1.36T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 + 3.08T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−7.26058981618342951598938493035, −6.64300252045573897227685997967, −6.21204596899562961230553248022, −5.53542520525175344202591196693, −4.75232904080527185088437523496, −4.20396291716412772201040066009, −3.57862782909946700058056215517, −2.33813419254822556875654056198, −1.79014450543595923552451435921, 0,
1.79014450543595923552451435921, 2.33813419254822556875654056198, 3.57862782909946700058056215517, 4.20396291716412772201040066009, 4.75232904080527185088437523496, 5.53542520525175344202591196693, 6.21204596899562961230553248022, 6.64300252045573897227685997967, 7.26058981618342951598938493035