L(s) = 1 | + (−0.900 − 1.43i)2-s + (0.974 − 0.222i)3-s + (−0.810 + 1.68i)4-s + (−1.19 − 1.19i)6-s + (1.46 − 0.164i)8-s + (0.900 − 0.433i)9-s + (−0.415 + 1.82i)12-s + (0.347 − 0.277i)13-s + (−0.387 − 0.485i)16-s + (−1.43 − 0.900i)18-s + (0.974 − 0.222i)23-s + (1.38 − 0.485i)24-s + (−0.900 − 0.433i)25-s + (−0.711 − 0.248i)26-s + (0.781 − 0.623i)27-s + ⋯ |
L(s) = 1 | + (−0.900 − 1.43i)2-s + (0.974 − 0.222i)3-s + (−0.810 + 1.68i)4-s + (−1.19 − 1.19i)6-s + (1.46 − 0.164i)8-s + (0.900 − 0.433i)9-s + (−0.415 + 1.82i)12-s + (0.347 − 0.277i)13-s + (−0.387 − 0.485i)16-s + (−1.43 − 0.900i)18-s + (0.974 − 0.222i)23-s + (1.38 − 0.485i)24-s + (−0.900 − 0.433i)25-s + (−0.711 − 0.248i)26-s + (0.781 − 0.623i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.014022319\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.014022319\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.974 + 0.222i)T \) |
| 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
−9.034482791200691377992584534995, −8.660342816148495825957294149486, −7.88139673977076624660903912216, −7.16824785216302396996083559641, −6.00738082805637352622339639953, −4.54465086309792356039472470332, −3.66536308297317511316081981009, −2.91094564209163515288697300508, −2.10026283813814153116905633521, −1.04591744591398606015232252720,
1.38548675979364395157753701400, 2.81714908532823150658520983667, 4.00386050507714591078603173112, 4.95577255689630978394102725256, 5.89503932554467345387337060847, 6.75098360553972472977521807623, 7.43906110709513718285693864166, 8.066174363673156350058764683868, 8.703747417080700604326694857743, 9.354375159977853069867328053516