L(s) = 1 | − 0.244·2-s − 1.94·4-s − 4.22·5-s + 0.0748·7-s + 0.962·8-s + 1.03·10-s − 3.64·11-s + 6.36·13-s − 0.0182·14-s + 3.64·16-s + 2.35·17-s + 1.06·19-s + 8.19·20-s + 0.890·22-s + 1.33·23-s + 12.8·25-s − 1.55·26-s − 0.145·28-s + 8.25·29-s − 7.90·31-s − 2.81·32-s − 0.575·34-s − 0.316·35-s − 2.87·37-s − 0.260·38-s − 4.06·40-s − 7.12·41-s + ⋯ |
L(s) = 1 | − 0.172·2-s − 0.970·4-s − 1.88·5-s + 0.0283·7-s + 0.340·8-s + 0.326·10-s − 1.09·11-s + 1.76·13-s − 0.00489·14-s + 0.911·16-s + 0.571·17-s + 0.244·19-s + 1.83·20-s + 0.189·22-s + 0.277·23-s + 2.56·25-s − 0.304·26-s − 0.0274·28-s + 1.53·29-s − 1.41·31-s − 0.497·32-s − 0.0987·34-s − 0.0534·35-s − 0.473·37-s − 0.0423·38-s − 0.643·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + T \) |
good | 2 | \( 1 + 0.244T + 2T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 - 0.0748T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 29 | \( 1 - 8.25T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 6.29T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 4.26T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 6.66T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 9.62T + 97T^{2} \) |
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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
−8.448665523605696026510354738186, −8.086204576228236573784076484569, −7.49934697276200480691016111139, −6.41444169249458488682101728752, −5.22037724088838374119724350948, −4.62403271044885205781805159092, −3.59736782404196480027171043532, −3.27506148500604090212427833217, −1.14478065706021540697446349490, 0,
1.14478065706021540697446349490, 3.27506148500604090212427833217, 3.59736782404196480027171043532, 4.62403271044885205781805159092, 5.22037724088838374119724350948, 6.41444169249458488682101728752, 7.49934697276200480691016111139, 8.086204576228236573784076484569, 8.448665523605696026510354738186