L(s) = 1 | + (0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s + (−1.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 25-s + (−0.499 + 0.866i)28-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s − 1.73i·52-s + (−1.5 − 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)79-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s + (−1.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 25-s + (−0.499 + 0.866i)28-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s − 1.73i·52-s + (−1.5 − 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9991992784\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9991992784\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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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
−11.19081008982845572032519831711, −10.27983054090362350502093924336, −9.209220304936024096435428160395, −8.277944232036800925987339716789, −7.64019389229716329241973809615, −6.69128923075655805848447186847, −5.50162721548453889247991148361, −4.54285279579294804562077956326, −3.05807907416143990652859176800, −2.24789893537749924278778897919,
1.43296185450519497176035749731, 2.78416591208739725714525265595, 4.52237044167214234497020971252, 5.08413684885488348875116890251, 6.54636895971164313390089867187, 7.04678911226768376231374881935, 8.076925676424842283292863589501, 9.318359714765459824120509206213, 10.10559421326159874727307151778, 10.71936811770433583140998684879