L(s) = 1 | − 1.23·3-s + 3.23·5-s − 1.47·9-s − 6.47·11-s + 0.763·13-s − 4.00·15-s − 4.47·17-s + 1.23·19-s − 4·23-s + 5.47·25-s + 5.52·27-s + 4.47·29-s − 2.47·31-s + 8.00·33-s + 4.47·37-s − 0.944·39-s + 8.47·41-s + 6.47·43-s − 4.76·45-s + 10.4·47-s + 5.52·51-s + 10·53-s − 20.9·55-s − 1.52·57-s + 9.23·59-s + 11.2·61-s + 2.47·65-s + ⋯ |
L(s) = 1 | − 0.713·3-s + 1.44·5-s − 0.490·9-s − 1.95·11-s + 0.211·13-s − 1.03·15-s − 1.08·17-s + 0.283·19-s − 0.834·23-s + 1.09·25-s + 1.06·27-s + 0.830·29-s − 0.444·31-s + 1.39·33-s + 0.735·37-s − 0.151·39-s + 1.32·41-s + 0.986·43-s − 0.710·45-s + 1.52·47-s + 0.774·51-s + 1.37·53-s − 2.82·55-s − 0.202·57-s + 1.20·59-s + 1.43·61-s + 0.306·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.392274113\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.392274113\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.720158930882663131053464858152, −7.994647389976667672137548719805, −7.03468266134560376851255626323, −6.17319485261003624198541789241, −5.58839402132175013148296073618, −5.24099471706668526143146906333, −4.17184093824631510286201721847, −2.52385360263673289479409211698, −2.40335105737748243788821728374, −0.70969926236305266170088297343,
0.70969926236305266170088297343, 2.40335105737748243788821728374, 2.52385360263673289479409211698, 4.17184093824631510286201721847, 5.24099471706668526143146906333, 5.58839402132175013148296073618, 6.17319485261003624198541789241, 7.03468266134560376851255626323, 7.994647389976667672137548719805, 8.720158930882663131053464858152