| L(s) = 1 | + 2.44·3-s + 3.46·5-s − 2.82·7-s + 2.99·9-s + 2.44·11-s − 3.46·13-s + 8.48·15-s + 4·17-s − 2.44·19-s − 6.92·21-s − 2.82·23-s + 6.99·25-s + 3.46·29-s − 5.65·31-s + 5.99·33-s − 9.79·35-s − 3.46·37-s − 8.48·39-s − 2·41-s − 12.2·43-s + 10.3·45-s + 11.3·47-s + 1.00·49-s + 9.79·51-s + 10.3·53-s + 8.48·55-s − 5.99·57-s + ⋯ |
| L(s) = 1 | + 1.41·3-s + 1.54·5-s − 1.06·7-s + 0.999·9-s + 0.738·11-s − 0.960·13-s + 2.19·15-s + 0.970·17-s − 0.561·19-s − 1.51·21-s − 0.589·23-s + 1.39·25-s + 0.643·29-s − 1.01·31-s + 1.04·33-s − 1.65·35-s − 0.569·37-s − 1.35·39-s − 0.312·41-s − 1.86·43-s + 1.54·45-s + 1.65·47-s + 0.142·49-s + 1.37·51-s + 1.42·53-s + 1.14·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.534947326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.534947326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 7.34T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−10.31992361653377872383792709097, −9.825260657546260206011991052466, −9.260654750296830573655221239834, −8.486826613080997835588144836994, −7.23924539786990797932610680634, −6.38453319740430877158589533634, −5.37422677875302419227534505818, −3.78347127923126069707803381458, −2.78458236989020716631294433563, −1.84922114757718931900306945304,
1.84922114757718931900306945304, 2.78458236989020716631294433563, 3.78347127923126069707803381458, 5.37422677875302419227534505818, 6.38453319740430877158589533634, 7.23924539786990797932610680634, 8.486826613080997835588144836994, 9.260654750296830573655221239834, 9.825260657546260206011991052466, 10.31992361653377872383792709097
