| L(s) = 1 | − 1.23·3-s − 3.23·5-s + 7-s − 1.47·9-s − 6.47·13-s + 4.00·15-s − 2.76·17-s − 4.47·19-s − 1.23·21-s + 23-s + 5.47·25-s + 5.52·27-s − 2·29-s − 3.23·31-s − 3.23·35-s + 6·37-s + 8.00·39-s − 10·41-s − 2.47·43-s + 4.76·45-s + 11.2·47-s + 49-s + 3.41·51-s − 6·53-s + 5.52·57-s − 6.76·59-s − 1.70·61-s + ⋯ |
| L(s) = 1 | − 0.713·3-s − 1.44·5-s + 0.377·7-s − 0.490·9-s − 1.79·13-s + 1.03·15-s − 0.670·17-s − 1.02·19-s − 0.269·21-s + 0.208·23-s + 1.09·25-s + 1.06·27-s − 0.371·29-s − 0.581·31-s − 0.546·35-s + 0.986·37-s + 1.28·39-s − 1.56·41-s − 0.376·43-s + 0.710·45-s + 1.63·47-s + 0.142·49-s + 0.478·51-s − 0.824·53-s + 0.732·57-s − 0.880·59-s − 0.218·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3393495781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3393495781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.728564544479102779404254930209, −8.067544426112568058260077228405, −7.34997251045170054463138988963, −6.73636062459097562181940985249, −5.70441769753123969515496104333, −4.77469103201578406647576399968, −4.37843766353409222720728412113, −3.22053234298976490583759322116, −2.17080862749283919522329467744, −0.35717431123399327097161442177,
0.35717431123399327097161442177, 2.17080862749283919522329467744, 3.22053234298976490583759322116, 4.37843766353409222720728412113, 4.77469103201578406647576399968, 5.70441769753123969515496104333, 6.73636062459097562181940985249, 7.34997251045170054463138988963, 8.067544426112568058260077228405, 8.728564544479102779404254930209
