L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + (−0.707 − 1.22i)19-s + (−0.5 + 0.866i)25-s + (−0.499 + 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)38-s + 1.41·41-s + 0.999·50-s + (0.707 − 1.22i)59-s + 0.999·64-s + (1 − 1.73i)67-s + (0.707 + 1.22i)68-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + (−0.707 − 1.22i)19-s + (−0.5 + 0.866i)25-s + (−0.499 + 0.866i)32-s − 1.41·34-s + (−0.707 + 1.22i)38-s + 1.41·41-s + 0.999·50-s + (0.707 − 1.22i)59-s + 0.999·64-s + (1 − 1.73i)67-s + (0.707 + 1.22i)68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8554519881\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8554519881\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−8.786730798574142798841957239021, −7.82263948952795508381687594057, −7.34561667366881049636912336533, −6.45279465060330201168052791981, −5.27352421058608895990154443443, −4.63522485560747612613519669813, −3.65534635771493383146997571848, −2.85220503842300141855344897917, −1.99380360492751140790703285030, −0.67591441734767280284797769524,
1.21473151227846289722687204000, 2.29823547647708984712685316879, 3.82458300370464019627451903443, 4.34608589508292863984545091000, 5.60627014802211170006788264246, 5.92067638445988745052820920810, 6.74912973938438901991341209043, 7.59161222418283715977337260793, 8.259598995173777107442554087910, 8.640323968566846055052344707089