L(s) = 1 | + (1.37 + 0.321i)2-s + (1.79 + 0.884i)4-s − 4.05·7-s + (2.18 + 1.79i)8-s + 0.985i·11-s + 4.94i·13-s + (−5.58 − 1.30i)14-s + (2.43 + 3.17i)16-s − 4.52·17-s + 2.60i·19-s + (−0.316 + 1.35i)22-s − 3.53·23-s + (−1.58 + 6.81i)26-s + (−7.27 − 3.58i)28-s − 7.59i·29-s + ⋯ |
L(s) = 1 | + (0.973 + 0.227i)2-s + (0.896 + 0.442i)4-s − 1.53·7-s + (0.773 + 0.634i)8-s + 0.297i·11-s + 1.37i·13-s + (−1.49 − 0.348i)14-s + (0.608 + 0.793i)16-s − 1.09·17-s + 0.597i·19-s + (−0.0674 + 0.289i)22-s − 0.737·23-s + (−0.311 + 1.33i)26-s + (−1.37 − 0.678i)28-s − 1.41i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.721659531\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.721659531\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.321i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 - 0.985iT - 11T^{2} \) |
| 13 | \( 1 - 4.94iT - 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 - 2.60iT - 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 + 7.59iT - 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 - 0.945iT - 37T^{2} \) |
| 41 | \( 1 + 0.568T + 41T^{2} \) |
| 43 | \( 1 - 8.45iT - 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 + 0.229iT - 53T^{2} \) |
| 59 | \( 1 - 9.10iT - 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 8.45iT - 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 3.28T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
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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
−9.628142462620334957099357066810, −8.831159104285512577507582704358, −7.75862222580095164318136515094, −6.86435275684740857587340255437, −6.39445675954794782733022676814, −5.73080278359888408522475146259, −4.39260150676998716790120674134, −3.97362470939901662592867168616, −2.85882354542373311998646347611, −1.93686500385048564617691828175,
0.41068102322151385237942777956, 2.18319868654231434153384664046, 3.20464490029619598930514802927, 3.67127914887873270509035782680, 4.92762050222265132515519990611, 5.68181222667056821065452792905, 6.51321022759325760489673748641, 7.00356079410466210260595042678, 8.081766882962070865699607609755, 9.151563710366805988537123792452