L(s) = 1 | − 0.546·3-s − 2.70·9-s − 1.70·11-s + 0.546·13-s − 3.11·17-s − 4.75·19-s − 5.40·23-s + 3.11·27-s − 0.298·29-s + 7.32·31-s + 0.929·33-s − 5.40·37-s − 0.298·39-s − 7.32·41-s + 4·43-s − 8.25·47-s + 1.70·51-s − 1.40·53-s + 2.59·57-s − 7.70·59-s + 9.89·61-s + 8·67-s + 2.95·69-s − 5.40·71-s + 11.3·73-s + 11.1·79-s + 6.40·81-s + ⋯ |
L(s) = 1 | − 0.315·3-s − 0.900·9-s − 0.513·11-s + 0.151·13-s − 0.755·17-s − 1.09·19-s − 1.12·23-s + 0.599·27-s − 0.0554·29-s + 1.31·31-s + 0.161·33-s − 0.888·37-s − 0.0477·39-s − 1.14·41-s + 0.609·43-s − 1.20·47-s + 0.238·51-s − 0.192·53-s + 0.343·57-s − 1.00·59-s + 1.26·61-s + 0.977·67-s + 0.355·69-s − 0.641·71-s + 1.33·73-s + 1.24·79-s + 0.711·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7964687241\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7964687241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.546T + 3T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 0.546T + 13T^{2} \) |
| 17 | \( 1 + 3.11T + 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 + 5.40T + 23T^{2} \) |
| 29 | \( 1 + 0.298T + 29T^{2} \) |
| 31 | \( 1 - 7.32T + 31T^{2} \) |
| 37 | \( 1 + 5.40T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8.25T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−7.80464314329343738987091055915, −6.74412395629802883828062070052, −6.38567730948970172874444761523, −5.65155653337521157677984721116, −4.95711658494598624678947483070, −4.27541290398350607731316516666, −3.39797257562260009540463345476, −2.54711345054479344072743172033, −1.83225397888700437354988471672, −0.40876501115478679853643355504,
0.40876501115478679853643355504, 1.83225397888700437354988471672, 2.54711345054479344072743172033, 3.39797257562260009540463345476, 4.27541290398350607731316516666, 4.95711658494598624678947483070, 5.65155653337521157677984721116, 6.38567730948970172874444761523, 6.74412395629802883828062070052, 7.80464314329343738987091055915