| L(s) = 1 | − 1.79·5-s − 7-s − 2.96·11-s + 6.95·13-s + 6.11·17-s + 0.967·19-s − 7.91·23-s − 1.76·25-s − 0.798·29-s + 0.798·31-s + 1.79·35-s + 6.76·37-s − 3.79·41-s − 9.71·43-s + 11.0·47-s + 49-s − 13.6·53-s + 5.33·55-s − 4.59·59-s + 8.56·61-s − 12.4·65-s + 13.5·67-s + 11.9·71-s + 4.33·73-s + 2.96·77-s − 17.6·79-s + 0.765·83-s + ⋯ |
| L(s) = 1 | − 0.804·5-s − 0.377·7-s − 0.894·11-s + 1.92·13-s + 1.48·17-s + 0.221·19-s − 1.65·23-s − 0.353·25-s − 0.148·29-s + 0.143·31-s + 0.303·35-s + 1.11·37-s − 0.593·41-s − 1.48·43-s + 1.61·47-s + 0.142·49-s − 1.87·53-s + 0.719·55-s − 0.598·59-s + 1.09·61-s − 1.55·65-s + 1.65·67-s + 1.41·71-s + 0.507·73-s + 0.338·77-s − 1.98·79-s + 0.0840·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.442563094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442563094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 1.79T + 5T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 - 6.95T + 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 - 0.967T + 19T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 + 0.798T + 29T^{2} \) |
| 31 | \( 1 - 0.798T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 + 3.79T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 4.33T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 - 0.765T + 83T^{2} \) |
| 89 | \( 1 + 3.98T + 89T^{2} \) |
| 97 | \( 1 + 6.86T + 97T^{2} \) |
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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
−8.148776880628017220387053699672, −7.58828369725174772284445223501, −6.60213040886150234611025562198, −5.89680111871518943254926814025, −5.37660692987237350587053113195, −4.20239713594692921139778733437, −3.64602330079433473095398389231, −3.03050387615154935024731562973, −1.76735063092266528693842950308, −0.63630450033341741494164250736,
0.63630450033341741494164250736, 1.76735063092266528693842950308, 3.03050387615154935024731562973, 3.64602330079433473095398389231, 4.20239713594692921139778733437, 5.37660692987237350587053113195, 5.89680111871518943254926814025, 6.60213040886150234611025562198, 7.58828369725174772284445223501, 8.148776880628017220387053699672
