L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 5·13-s + 14-s + 16-s + 17-s + 6·19-s − 20-s + 8·23-s − 4·25-s − 5·26-s − 28-s + 4·29-s − 6·31-s − 32-s − 34-s + 35-s − 5·37-s − 6·38-s + 40-s − 4·41-s − 43-s − 8·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 0.223·20-s + 1.66·23-s − 4/5·25-s − 0.980·26-s − 0.188·28-s + 0.742·29-s − 1.07·31-s − 0.176·32-s − 0.171·34-s + 0.169·35-s − 0.821·37-s − 0.973·38-s + 0.158·40-s − 0.624·41-s − 0.152·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.20657878809797, −12.45827756772504, −12.01593236108879, −11.56262889863525, −11.22714398136494, −10.69472916041604, −10.34811663495117, −9.673293341993755, −9.402607714064607, −8.683117463849236, −8.614682904321318, −7.903422943911572, −7.471236380415979, −6.890300404505820, −6.718973159295719, −5.835117503306879, −5.590373343349456, −4.975952926657628, −4.271247847916635, −3.521139192412973, −3.346080413250586, −2.799708243613053, −1.897590695660335, −1.290763244112132, −0.8315606916500988, 0,
0.8315606916500988, 1.290763244112132, 1.897590695660335, 2.799708243613053, 3.346080413250586, 3.521139192412973, 4.271247847916635, 4.975952926657628, 5.590373343349456, 5.835117503306879, 6.718973159295719, 6.890300404505820, 7.471236380415979, 7.903422943911572, 8.614682904321318, 8.683117463849236, 9.402607714064607, 9.673293341993755, 10.34811663495117, 10.69472916041604, 11.22714398136494, 11.56262889863525, 12.01593236108879, 12.45827756772504, 13.20657878809797