L(s) = 1 | + (−0.294 − 1.38i)2-s + (−0.913 + 0.406i)4-s + (−0.978 − 0.207i)5-s + (−1.40 − 0.147i)7-s + 1.41i·10-s + (0.209 + 1.98i)14-s + (−0.669 + 0.743i)16-s + (0.978 − 0.207i)20-s + (1.34 − 0.437i)28-s + (−1.40 − 0.147i)29-s + (0.669 + 0.743i)31-s + (1.22 + 0.707i)32-s + (1.34 + 0.437i)35-s + (0.809 + 0.587i)37-s + (0.913 + 0.406i)47-s + ⋯ |
L(s) = 1 | + (−0.294 − 1.38i)2-s + (−0.913 + 0.406i)4-s + (−0.978 − 0.207i)5-s + (−1.40 − 0.147i)7-s + 1.41i·10-s + (0.209 + 1.98i)14-s + (−0.669 + 0.743i)16-s + (0.978 − 0.207i)20-s + (1.34 − 0.437i)28-s + (−1.40 − 0.147i)29-s + (0.669 + 0.743i)31-s + (1.22 + 0.707i)32-s + (1.34 + 0.437i)35-s + (0.809 + 0.587i)37-s + (0.913 + 0.406i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2865476273\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2865476273\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.294 + 1.38i)T + (-0.913 + 0.406i)T^{2} \) |
| 5 | \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 7 | \( 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2} \) |
| 13 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2} \) |
| 31 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 0.946i)T + (0.104 + 0.994i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 83 | \( 1 + (1.05 + 0.946i)T + (0.104 + 0.994i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \) |
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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
−9.168033001358448871538273304979, −8.278272377611523258502249739376, −7.44760574657096182683443564092, −6.62890295781090548561181999751, −5.86057941768225048014868721627, −4.52889055637638019982745615063, −3.81833264757722944984335636457, −3.23443198407283340209812329738, −2.41182617866528822862705582463, −1.02230226057611068899110721824,
0.22561377763466394589823670952, 2.46223409037479399913587667835, 3.47325986015118762706693746385, 4.18842778437837025498472756857, 5.37321767324087286826291875161, 6.01255892558143738074255555127, 6.74361957997649425743729601284, 7.26503276429444936309429349663, 7.944644527540135198138272106939, 8.565973005058522835196151500526