| L(s) = 1 | + 3.46·5-s − 3.46·11-s − 2·13-s − 3.46·17-s − 4·19-s + 3.46·23-s + 6.99·25-s − 4·31-s + 2·37-s − 10.3·41-s + 4·43-s + 6.92·47-s − 6.92·53-s − 11.9·55-s − 6.92·59-s + 10·61-s − 6.92·65-s + 4·67-s + 10.3·71-s − 14·73-s − 8·79-s − 11.9·85-s + 3.46·89-s − 13.8·95-s − 14·97-s − 3.46·101-s − 4·103-s + ⋯ |
| L(s) = 1 | + 1.54·5-s − 1.04·11-s − 0.554·13-s − 0.840·17-s − 0.917·19-s + 0.722·23-s + 1.39·25-s − 0.718·31-s + 0.328·37-s − 1.62·41-s + 0.609·43-s + 1.01·47-s − 0.951·53-s − 1.61·55-s − 0.901·59-s + 1.28·61-s − 0.859·65-s + 0.488·67-s + 1.23·71-s − 1.63·73-s − 0.900·79-s − 1.30·85-s + 0.367·89-s − 1.42·95-s − 1.42·97-s − 0.344·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.51218116529908702620620166504, −6.77167896444280890855160561782, −6.21512305737288936287229642991, −5.37006982049676306500307958175, −5.02114905098027128279915958922, −4.05548690470664349380117502531, −2.79489885101287138119063207250, −2.34188766022526201091472767243, −1.50682940868252350405565775611, 0,
1.50682940868252350405565775611, 2.34188766022526201091472767243, 2.79489885101287138119063207250, 4.05548690470664349380117502531, 5.02114905098027128279915958922, 5.37006982049676306500307958175, 6.21512305737288936287229642991, 6.77167896444280890855160561782, 7.51218116529908702620620166504
