| L(s) = 1 | + 3-s − 7-s + 9-s + 4·13-s + 2·17-s + 6·19-s − 21-s + 4·23-s + 27-s + 4·29-s − 6·37-s + 4·39-s − 8·43-s + 4·47-s + 49-s + 2·51-s + 6·53-s + 6·57-s + 4·59-s − 6·61-s − 63-s + 4·67-s + 4·69-s + 6·79-s + 81-s + 14·83-s + 4·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.485·17-s + 1.37·19-s − 0.218·21-s + 0.834·23-s + 0.192·27-s + 0.742·29-s − 0.986·37-s + 0.640·39-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.794·57-s + 0.520·59-s − 0.768·61-s − 0.125·63-s + 0.488·67-s + 0.481·69-s + 0.675·79-s + 1/9·81-s + 1.53·83-s + 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.742869824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.742869824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.06735916583434, −12.26985102360225, −12.00999659599116, −11.52310261010661, −10.92742695565397, −10.44867823820170, −10.09289874563120, −9.471777238347937, −9.183135049937335, −8.519424792693972, −8.360344689352186, −7.640484818059170, −7.180337870186377, −6.816004922102749, −6.136292757231744, −5.764486227495435, −5.004914911714742, −4.789467071083918, −3.826727652413143, −3.454920235903734, −3.176903323373761, −2.463274744549007, −1.773867375078475, −1.093717491839979, −0.6488309615320283,
0.6488309615320283, 1.093717491839979, 1.773867375078475, 2.463274744549007, 3.176903323373761, 3.454920235903734, 3.826727652413143, 4.789467071083918, 5.004914911714742, 5.764486227495435, 6.136292757231744, 6.816004922102749, 7.180337870186377, 7.640484818059170, 8.360344689352186, 8.519424792693972, 9.183135049937335, 9.471777238347937, 10.09289874563120, 10.44867823820170, 10.92742695565397, 11.52310261010661, 12.00999659599116, 12.26985102360225, 13.06735916583434
