L(s) = 1 | + (0.900 + 1.43i)2-s + (0.623 + 0.781i)3-s + (−0.810 + 1.68i)4-s + (−0.559 + 1.59i)6-s + (−1.46 + 0.164i)8-s + (−0.222 + 0.974i)9-s + (−1.82 + 0.415i)12-s + (0.347 − 0.277i)13-s + (−0.387 − 0.485i)16-s + (−1.59 + 0.559i)18-s + (−0.974 + 0.222i)23-s + (−1.03 − 1.03i)24-s + (−0.900 − 0.433i)25-s + (0.711 + 0.248i)26-s + (−0.900 + 0.433i)27-s + ⋯ |
L(s) = 1 | + (0.900 + 1.43i)2-s + (0.623 + 0.781i)3-s + (−0.810 + 1.68i)4-s + (−0.559 + 1.59i)6-s + (−1.46 + 0.164i)8-s + (−0.222 + 0.974i)9-s + (−1.82 + 0.415i)12-s + (0.347 − 0.277i)13-s + (−0.387 − 0.485i)16-s + (−1.59 + 0.559i)18-s + (−0.974 + 0.222i)23-s + (−1.03 − 1.03i)24-s + (−0.900 − 0.433i)25-s + (0.711 + 0.248i)26-s + (−0.900 + 0.433i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.141797210\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.141797210\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.623 - 0.781i)T \) |
| 23 | \( 1 + (0.974 - 0.222i)T \) |
| 29 | \( 1 + (-0.433 + 0.900i)T \) |
good | 2 | \( 1 + (-0.900 - 1.43i)T + (-0.433 + 0.900i)T^{2} \) |
| 5 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 13 | \( 1 + (-0.347 + 0.277i)T + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 31 | \( 1 + (-1.19 + 0.752i)T + (0.433 - 0.900i)T^{2} \) |
| 37 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 41 | \( 1 + (0.158 + 0.158i)T + iT^{2} \) |
| 43 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 47 | \( 1 + (0.0739 - 0.656i)T + (-0.974 - 0.222i)T^{2} \) |
| 53 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 + 1.80iT - T^{2} \) |
| 61 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 67 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 71 | \( 1 + (-1.21 - 1.52i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (0.559 + 0.351i)T + (0.433 + 0.900i)T^{2} \) |
| 79 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 83 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
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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
−9.650239061924131414170486370264, −8.488712764208218614405356308111, −8.130984573410924484093652306194, −7.45213977845549572814994248294, −6.35379435364425944430060581725, −5.78880222651920852442435500629, −4.89093088620405652667517717260, −4.15430059526751539138538093864, −3.54168703298228365615105393226, −2.33713761154925118500050241129,
1.20406104806879388650302671359, 2.09124909154696315457701405354, 2.97453377920416241530484285525, 3.74356672875853793723736063057, 4.56067290256980701126197138431, 5.65739466991993815742052384138, 6.43157222350187737731536063157, 7.42372256683071457671124364597, 8.342454853377084904571170299494, 9.104854719991092399194593604001