L(s) = 1 | − i·2-s + 2.73i·3-s − 4-s + 2.73·6-s − 1.26i·7-s + i·8-s − 4.46·9-s + 1.46·11-s − 2.73i·12-s − 1.46i·13-s − 1.26·14-s + 16-s − 1.46i·17-s + 4.46i·18-s + 4.19·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.57i·3-s − 0.5·4-s + 1.11·6-s − 0.479i·7-s + 0.353i·8-s − 1.48·9-s + 0.441·11-s − 0.788i·12-s − 0.406i·13-s − 0.338·14-s + 0.250·16-s − 0.355i·17-s + 1.05i·18-s + 0.962·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.558390198\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.558390198\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 - iT \) |
good | 3 | \( 1 - 2.73iT - 3T^{2} \) |
| 7 | \( 1 + 1.26iT - 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 + 1.46iT - 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 + 1.26iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 0.196T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 13.6iT - 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 5.26T + 79T^{2} \) |
| 83 | \( 1 - 5.26iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.513928421384187907222737444308, −9.036775485302498458522646529376, −8.055299798404343624482566501155, −7.11118614632229563695409027315, −5.78483252997752371096327570559, −5.07231044088994821649847385769, −4.30080619409569732522080765437, −3.53839405809838761439950205447, −2.86820236731229023909658896955, −1.15404701267860452455985041143,
0.68751852305849781152909956136, 1.88321315829624607037759863753, 2.93516964188152706749074709112, 4.28932651868748553973147135274, 5.38224302748975609646192112287, 6.18629492953428028726422166611, 6.80170566826505063176265144001, 7.31124034696581311298476613016, 8.400400058292028307658538958972, 8.570400256540429979728589091938