| L(s) = 1 | − 3-s + 3.27·5-s − 4·7-s + 9-s + 5.27·11-s − 3.27·13-s − 3.27·15-s + 17-s + 5.27·19-s + 4·21-s + 5.27·23-s + 5.72·25-s − 27-s + 2·29-s − 6.54·31-s − 5.27·33-s − 13.0·35-s + 4.54·37-s + 3.27·39-s + 0.725·41-s + 2.72·43-s + 3.27·45-s + 10.5·47-s + 9·49-s − 51-s − 10·53-s + 17.2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.46·5-s − 1.51·7-s + 0.333·9-s + 1.59·11-s − 0.908·13-s − 0.845·15-s + 0.242·17-s + 1.21·19-s + 0.872·21-s + 1.09·23-s + 1.14·25-s − 0.192·27-s + 0.371·29-s − 1.17·31-s − 0.918·33-s − 2.21·35-s + 0.747·37-s + 0.524·39-s + 0.113·41-s + 0.415·43-s + 0.488·45-s + 1.53·47-s + 1.28·49-s − 0.140·51-s − 1.37·53-s + 2.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.541593530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541593530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 + 3.27T + 13T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.54T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 - 0.725T + 41T^{2} \) |
| 43 | \( 1 - 2.72T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 6.54T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 + 6.54T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.54T + 97T^{2} \) |
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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
−9.926750224915221156368641336581, −9.528789474420900999293700314389, −9.082335585514177024740186009524, −7.24653827370191413082663128304, −6.64976883088664731140621630891, −5.92545563859123327767736448886, −5.16968739069446684466032270925, −3.75376351078369794200565867084, −2.60285870910209392545653716411, −1.11513149017912433233685104146,
1.11513149017912433233685104146, 2.60285870910209392545653716411, 3.75376351078369794200565867084, 5.16968739069446684466032270925, 5.92545563859123327767736448886, 6.64976883088664731140621630891, 7.24653827370191413082663128304, 9.082335585514177024740186009524, 9.528789474420900999293700314389, 9.926750224915221156368641336581
