L(s) = 1 | + (−0.707 + 1.22i)5-s + 1.41·13-s + (0.707 + 1.22i)17-s + (−2 + 3.46i)23-s + (1.50 + 2.59i)25-s + 6·29-s + (−2.82 − 4.89i)31-s + (2 − 3.46i)37-s − 7.07·41-s + 4·43-s + (−5.65 + 9.79i)47-s + (−2 − 3.46i)53-s + (2.82 + 4.89i)59-s + (−2.12 + 3.67i)61-s + (−1.00 + 1.73i)65-s + ⋯ |
L(s) = 1 | + (−0.316 + 0.547i)5-s + 0.392·13-s + (0.171 + 0.297i)17-s + (−0.417 + 0.722i)23-s + (0.300 + 0.519i)25-s + 1.11·29-s + (−0.508 − 0.879i)31-s + (0.328 − 0.569i)37-s − 1.10·41-s + 0.609·43-s + (−0.825 + 1.42i)47-s + (−0.274 − 0.475i)53-s + (0.368 + 0.637i)59-s + (−0.271 + 0.470i)61-s + (−0.124 + 0.214i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.343556040\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.343556040\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.707 - 1.22i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2.82 + 4.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (5.65 - 9.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.82 - 4.89i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.12 - 3.67i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-3.53 - 6.12i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + (6.36 - 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.709955787917524289830757512472, −7.967091426738007642685878360430, −7.35972823160661148093084633647, −6.55139382511613056746464279755, −5.87424843833091500003629361379, −5.00197417155273096594892817286, −4.00508825526179851890853815716, −3.35063530308243992183259371680, −2.38258301681496374937143377112, −1.18508285253084945107038954329,
0.43935451576322650180955812765, 1.61021525472059948799498983903, 2.80278313507184333261277285481, 3.69628451057137604844129335012, 4.62836739168375141294393875991, 5.15792322718924534890114192508, 6.22893890494919855910068075631, 6.80255796981145566518973990908, 7.74395577729042118799256487329, 8.510419746777302725531048277360