L(s) = 1 | + i·2-s − 1.08i·3-s − 4-s + 1.08i·5-s + 1.08·6-s − i·8-s + 1.82·9-s − 1.08·10-s + (−1.41 − 3i)11-s + 1.08i·12-s + 2.29·13-s + 1.17·15-s + 16-s + 1.82i·18-s − 5.54·19-s − 1.08i·20-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.624i·3-s − 0.5·4-s + 0.484i·5-s + 0.441·6-s − 0.353i·8-s + 0.609·9-s − 0.342·10-s + (−0.426 − 0.904i)11-s + 0.312i·12-s + 0.636·13-s + 0.302·15-s + 0.250·16-s + 0.430i·18-s − 1.27·19-s − 0.242i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.587015274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.587015274\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.41 + 3i)T \) |
good | 3 | \( 1 + 1.08iT - 3T^{2} \) |
| 5 | \( 1 - 1.08iT - 5T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 1.75iT - 29T^{2} \) |
| 31 | \( 1 + 1.39iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 4.59T + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 - 4.90iT - 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 + 7.20iT - 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 3.56iT - 89T^{2} \) |
| 97 | \( 1 + 8.60iT - 97T^{2} \) |
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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
−9.793612545367514086572783198431, −8.779824253908352897297295442450, −8.141047055878656562820321289681, −7.28057939972844566659464079678, −6.55571217358634870123075991514, −5.95012341784612875278311110696, −4.78334061025514643307977781727, −3.72210776171302024877970478964, −2.49441296138584440191662549811, −0.884864412827685459142034235266,
1.19270590449682570131363329530, 2.48763245035909541536286516443, 3.75745542300058965709636147425, 4.57523202526214433644734184946, 5.14747272142092290461806656068, 6.49220980844134114286955641392, 7.48103294025445905329199330572, 8.577291415490005276697321231094, 9.171824366329973538055539686503, 9.966673819945883621841304462728