L(s) = 1 | − 0.702·2-s − 1.50·4-s + 2.41·5-s + 0.380·7-s + 2.46·8-s − 1.69·10-s − 11-s + 2.87·13-s − 0.267·14-s + 1.28·16-s − 1.85·17-s − 2.38·19-s − 3.63·20-s + 0.702·22-s + 5.66·23-s + 0.811·25-s − 2.01·26-s − 0.573·28-s − 2.60·29-s − 4.67·31-s − 5.82·32-s + 1.30·34-s + 0.917·35-s − 10.7·37-s + 1.67·38-s + 5.93·40-s + 0.412·41-s + ⋯ |
L(s) = 1 | − 0.496·2-s − 0.753·4-s + 1.07·5-s + 0.143·7-s + 0.871·8-s − 0.535·10-s − 0.301·11-s + 0.796·13-s − 0.0714·14-s + 0.320·16-s − 0.449·17-s − 0.547·19-s − 0.811·20-s + 0.149·22-s + 1.18·23-s + 0.162·25-s − 0.396·26-s − 0.108·28-s − 0.483·29-s − 0.840·31-s − 1.03·32-s + 0.223·34-s + 0.155·35-s − 1.77·37-s + 0.272·38-s + 0.939·40-s + 0.0644·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 0.702T + 2T^{2} \) |
| 5 | \( 1 - 2.41T + 5T^{2} \) |
| 7 | \( 1 - 0.380T + 7T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 0.412T + 41T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 67 | \( 1 - 8.09T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
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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
−7.967771325828566517493350119182, −6.97153440400839904546682953295, −6.38363974463652627314320244198, −5.36850020073741760447302474758, −5.09093658499672585995331582170, −4.05107041090242887136977607737, −3.24159689012746037759553255970, −2.00861459942636786867778963574, −1.36967888324310307301791779164, 0,
1.36967888324310307301791779164, 2.00861459942636786867778963574, 3.24159689012746037759553255970, 4.05107041090242887136977607737, 5.09093658499672585995331582170, 5.36850020073741760447302474758, 6.38363974463652627314320244198, 6.97153440400839904546682953295, 7.967771325828566517493350119182