L(s) = 1 | + (−1.82 − 0.205i)2-s + (0.0747 + 0.997i)3-s + (2.30 + 0.525i)4-s + (0.0685 − 1.83i)6-s + (−2.35 − 0.823i)8-s + (−0.988 + 0.149i)9-s + (−0.351 + 2.33i)12-s + (0.317 − 0.658i)13-s + (1.99 + 0.959i)16-s + (1.83 − 0.0685i)18-s + (−0.781 − 0.623i)23-s + (0.645 − 2.40i)24-s + (−0.222 + 0.974i)25-s + (−0.712 + 1.13i)26-s + (−0.222 − 0.974i)27-s + ⋯ |
L(s) = 1 | + (−1.82 − 0.205i)2-s + (0.0747 + 0.997i)3-s + (2.30 + 0.525i)4-s + (0.0685 − 1.83i)6-s + (−2.35 − 0.823i)8-s + (−0.988 + 0.149i)9-s + (−0.351 + 2.33i)12-s + (0.317 − 0.658i)13-s + (1.99 + 0.959i)16-s + (1.83 − 0.0685i)18-s + (−0.781 − 0.623i)23-s + (0.645 − 2.40i)24-s + (−0.222 + 0.974i)25-s + (−0.712 + 1.13i)26-s + (−0.222 − 0.974i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4049773576\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4049773576\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0747 - 0.997i)T \) |
| 23 | \( 1 + (0.781 + 0.623i)T \) |
| 29 | \( 1 + (-0.294 - 0.955i)T \) |
good | 2 | \( 1 + (1.82 + 0.205i)T + (0.974 + 0.222i)T^{2} \) |
| 5 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 13 | \( 1 + (-0.317 + 0.658i)T + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 31 | \( 1 + (0.216 - 1.91i)T + (-0.974 - 0.222i)T^{2} \) |
| 37 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 41 | \( 1 + (-1.38 - 1.38i)T + iT^{2} \) |
| 43 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 47 | \( 1 + (-0.584 - 1.66i)T + (-0.781 + 0.623i)T^{2} \) |
| 53 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 + 0.445iT - T^{2} \) |
| 61 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 67 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 71 | \( 1 + (0.268 + 0.129i)T + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.00837 + 0.0743i)T + (-0.974 + 0.222i)T^{2} \) |
| 79 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 83 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 97 | \( 1 + (0.433 - 0.900i)T^{2} \) |
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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
−9.463363651812825403125324909826, −9.003805542940482517652783306906, −8.230603444815428217545675403472, −7.69711246208561049971270705276, −6.63995539877719582092083007278, −5.81833068311709813141475904699, −4.71377084984173116892604850396, −3.40304240840418718260915861266, −2.70585673477413269350367147354, −1.32291495589009466147394620293,
0.54848036853844730473019349729, 1.88205690466628513365784688854, 2.44829314163191406724526771796, 3.97412916633236516571457341200, 5.76846075532926637792861393009, 6.23347382357981269076510613006, 7.09959847354561783315851377374, 7.69510776160540458941528131670, 8.296241153703000780475872172151, 8.972174275315096478549266124025