L(s) = 1 | − 2-s − 1.25·3-s + 4-s + 5-s + 1.25·6-s + 1.88·7-s − 8-s − 1.41·9-s − 10-s − 3.48·11-s − 1.25·12-s − 6.95·13-s − 1.88·14-s − 1.25·15-s + 16-s + 2.07·17-s + 1.41·18-s + 3.03·19-s + 20-s − 2.37·21-s + 3.48·22-s + 6.09·23-s + 1.25·24-s + 25-s + 6.95·26-s + 5.55·27-s + 1.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.726·3-s + 0.5·4-s + 0.447·5-s + 0.513·6-s + 0.713·7-s − 0.353·8-s − 0.471·9-s − 0.316·10-s − 1.05·11-s − 0.363·12-s − 1.92·13-s − 0.504·14-s − 0.324·15-s + 0.250·16-s + 0.504·17-s + 0.333·18-s + 0.696·19-s + 0.223·20-s − 0.518·21-s + 0.743·22-s + 1.27·23-s + 0.256·24-s + 0.200·25-s + 1.36·26-s + 1.06·27-s + 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 1.25T + 3T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 6.95T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 1.90T + 29T^{2} \) |
| 31 | \( 1 - 8.71T + 31T^{2} \) |
| 37 | \( 1 + 7.53T + 37T^{2} \) |
| 41 | \( 1 - 2.87T + 41T^{2} \) |
| 43 | \( 1 - 8.26T + 43T^{2} \) |
| 47 | \( 1 + 0.387T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 - 4.66T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 + 3.66T + 97T^{2} \) |
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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
−7.896103193234611321490772479364, −7.53183385913984851296504934511, −6.72953518168572452345191178947, −5.74532347832794512945696144455, −5.15851174181380552301826152910, −4.71856471765542492231992833624, −2.95551592826889692691714255862, −2.47730620088101945932137289346, −1.18638041785933017448880353192, 0,
1.18638041785933017448880353192, 2.47730620088101945932137289346, 2.95551592826889692691714255862, 4.71856471765542492231992833624, 5.15851174181380552301826152910, 5.74532347832794512945696144455, 6.72953518168572452345191178947, 7.53183385913984851296504934511, 7.896103193234611321490772479364