L(s) = 1 | + (−1.40 − 0.147i)2-s + (0.978 + 0.207i)4-s + (−0.104 − 0.994i)5-s + (1.05 + 0.946i)7-s + 1.41i·10-s + (−1.33 − 1.48i)14-s + (−0.913 − 0.406i)16-s + (0.104 − 0.994i)20-s + (0.831 + 1.14i)28-s + (1.05 + 0.946i)29-s + (0.913 − 0.406i)31-s + (1.22 + 0.707i)32-s + (0.831 − 1.14i)35-s + (−0.309 − 0.951i)37-s + (−0.978 + 0.207i)47-s + ⋯ |
L(s) = 1 | + (−1.40 − 0.147i)2-s + (0.978 + 0.207i)4-s + (−0.104 − 0.994i)5-s + (1.05 + 0.946i)7-s + 1.41i·10-s + (−1.33 − 1.48i)14-s + (−0.913 − 0.406i)16-s + (0.104 − 0.994i)20-s + (0.831 + 1.14i)28-s + (1.05 + 0.946i)29-s + (0.913 − 0.406i)31-s + (1.22 + 0.707i)32-s + (0.831 − 1.14i)35-s + (−0.309 − 0.951i)37-s + (−0.978 + 0.207i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6936895997\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6936895997\) |
\(L(1)\) |
|
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 | \( 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2} \) |
| 5 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 7 | \( 1 + (-1.05 - 0.946i)T + (0.104 + 0.994i)T^{2} \) |
| 13 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.05 - 0.946i)T + (0.104 + 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 61 | \( 1 + (0.575 - 1.29i)T + (-0.669 - 0.743i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 83 | \( 1 + (-0.575 + 1.29i)T + (-0.669 - 0.743i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
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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
−8.847206461156585378812767736345, −8.299383439206357500573099335958, −7.72967751741105135348428881627, −6.83355146401484688074501048152, −5.71573477101366633155197827730, −4.94536263413458648142353165170, −4.36877091084857588235113715557, −2.77575248989873258256927455131, −1.82524649508512336969972772767, −0.986024687277895092403832581013,
0.943320439875923627742846021233, 1.98695291646014666979973795395, 3.10906786774266572311217682694, 4.24552113599154057941209045877, 4.93944987967121737237706757668, 6.34626032333108163835004368328, 6.89330589959720105934783122240, 7.52190817222345719087511905847, 8.185654791155497985254466786459, 8.607276713924012461236296866615