| L(s) = 1 | + 1.41·3-s − 7-s − 0.999·9-s − 2.82·11-s + 2.82·13-s + 6·17-s + 4.24·19-s − 1.41·21-s − 5·25-s − 5.65·27-s − 1.41·29-s + 10·31-s − 4.00·33-s − 4.24·37-s + 4.00·39-s − 10·41-s + 8.48·43-s − 2·47-s + 49-s + 8.48·51-s + 12.7·53-s + 6·57-s + 1.41·59-s + 0.999·63-s − 11.3·67-s + 12·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.377·7-s − 0.333·9-s − 0.852·11-s + 0.784·13-s + 1.45·17-s + 0.973·19-s − 0.308·21-s − 25-s − 1.08·27-s − 0.262·29-s + 1.79·31-s − 0.696·33-s − 0.697·37-s + 0.640·39-s − 1.56·41-s + 1.29·43-s − 0.291·47-s + 0.142·49-s + 1.18·51-s + 1.74·53-s + 0.794·57-s + 0.184·59-s + 0.125·63-s − 1.38·67-s + 1.42·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.468110128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.468110128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.933362506897731648255144449857, −7.51030015923472088193581579154, −6.54867570244693370145708598956, −5.66510987231663151958406361164, −5.33960155151513364503447632186, −4.12604216291950117072435136127, −3.32347440036956222522007420993, −2.93914754800258505159060006496, −1.93890938618690512400894031641, −0.75983021637776817354262346560,
0.75983021637776817354262346560, 1.93890938618690512400894031641, 2.93914754800258505159060006496, 3.32347440036956222522007420993, 4.12604216291950117072435136127, 5.33960155151513364503447632186, 5.66510987231663151958406361164, 6.54867570244693370145708598956, 7.51030015923472088193581579154, 7.933362506897731648255144449857
