| L(s) = 1 | − 1.23·3-s + 4.47·7-s − 1.47·9-s − 3.23·13-s − 6.47·17-s + 19-s − 5.52·21-s + 2·23-s + 5.52·27-s + 2·29-s − 1.52·31-s + 4.76·37-s + 4.00·39-s + 3.52·41-s − 0.472·43-s + 12.4·47-s + 13.0·49-s + 8.00·51-s + 11.2·53-s − 1.23·57-s − 10.4·59-s − 4.47·61-s − 6.58·63-s + 1.23·67-s − 2.47·69-s − 1.52·71-s + 6.47·73-s + ⋯ |
| L(s) = 1 | − 0.713·3-s + 1.69·7-s − 0.490·9-s − 0.897·13-s − 1.56·17-s + 0.229·19-s − 1.20·21-s + 0.417·23-s + 1.06·27-s + 0.371·29-s − 0.274·31-s + 0.783·37-s + 0.640·39-s + 0.550·41-s − 0.0720·43-s + 1.81·47-s + 1.85·49-s + 1.12·51-s + 1.54·53-s − 0.163·57-s − 1.36·59-s − 0.572·61-s − 0.829·63-s + 0.151·67-s − 0.297·69-s − 0.181·71-s + 0.757·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.448693731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.448693731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 1.23T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 - 6.47T + 73T^{2} \) |
| 79 | \( 1 + 6.47T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 - 4.18T + 97T^{2} \) |
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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
−8.648551117302615527115396342505, −7.64005750182550139189269447279, −7.15228737477916643590786041528, −6.13580075056375641079024725939, −5.44159806472607436275098588855, −4.71696945197437841863755151464, −4.29346605372906466522512129151, −2.75296059343823700122785558885, −1.98647378972281012124670374091, −0.72063252434568800442681144632,
0.72063252434568800442681144632, 1.98647378972281012124670374091, 2.75296059343823700122785558885, 4.29346605372906466522512129151, 4.71696945197437841863755151464, 5.44159806472607436275098588855, 6.13580075056375641079024725939, 7.15228737477916643590786041528, 7.64005750182550139189269447279, 8.648551117302615527115396342505
