L(s) = 1 | + (−1.05 − 0.946i)2-s + (0.104 + 0.994i)4-s + (0.669 + 0.743i)5-s + (0.575 − 1.29i)7-s − 1.41i·10-s + (−1.82 + 0.813i)14-s + (0.978 − 0.207i)16-s + (−0.669 + 0.743i)20-s + (1.34 + 0.437i)28-s + (0.575 − 1.29i)29-s + (−0.978 − 0.207i)31-s + (−1.22 − 0.707i)32-s + (1.34 − 0.437i)35-s + (0.809 − 0.587i)37-s + (−0.104 + 0.994i)47-s + ⋯ |
L(s) = 1 | + (−1.05 − 0.946i)2-s + (0.104 + 0.994i)4-s + (0.669 + 0.743i)5-s + (0.575 − 1.29i)7-s − 1.41i·10-s + (−1.82 + 0.813i)14-s + (0.978 − 0.207i)16-s + (−0.669 + 0.743i)20-s + (1.34 + 0.437i)28-s + (0.575 − 1.29i)29-s + (−0.978 − 0.207i)31-s + (−1.22 − 0.707i)32-s + (1.34 − 0.437i)35-s + (0.809 − 0.587i)37-s + (−0.104 + 0.994i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8385158395\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8385158395\) |
\(L(1)\) |
|
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 | \( 1 + (1.05 + 0.946i)T + (0.104 + 0.994i)T^{2} \) |
| 5 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 7 | \( 1 + (-0.575 + 1.29i)T + (-0.669 - 0.743i)T^{2} \) |
| 13 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 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 | \( 1 + (-0.575 + 1.29i)T + (-0.669 - 0.743i)T^{2} \) |
| 31 | \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 61 | \( 1 + (-0.294 - 1.38i)T + (-0.913 + 0.406i)T^{2} \) |
| 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 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (0.294 + 1.38i)T + (-0.913 + 0.406i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.808625685508794493427708559991, −7.889996433344560142141658467181, −7.46971045709272223816740476681, −6.49796580862556547508359017279, −5.71748318297454160358940938034, −4.55768785846129787443908796992, −3.67247545829708983302722207434, −2.65020510312082488044075245792, −1.90015704873942300471707223656, −0.821829644646311025597775298674,
1.23867089066437498108644143062, 2.19209606933399459689158884847, 3.45038715937489672898463493549, 4.93590701241613357464966347281, 5.37012186674422056324369935003, 6.11260387944916303676526505838, 6.84353609735706319822650591388, 7.75143436673168176759806799549, 8.456575489219135199258142768811, 8.875209281309099304413383225257