| L(s) = 1 | − 1.61e5·3-s − 9.10e6·5-s + 4.58e8·7-s + 1.57e10·9-s − 1.21e11·11-s − 2.06e11·13-s + 1.47e12·15-s + 6.23e11·17-s + 1.95e13·19-s − 7.42e13·21-s + 1.83e14·23-s − 3.93e14·25-s − 8.59e14·27-s − 3.78e15·29-s − 6.71e15·31-s + 1.96e16·33-s − 4.17e15·35-s − 1.47e16·37-s + 3.33e16·39-s + 1.63e17·41-s + 9.69e16·43-s − 1.43e17·45-s + 1.37e17·47-s − 3.48e17·49-s − 1.01e17·51-s + 9.54e17·53-s + 1.10e18·55-s + ⋯ |
| L(s) = 1 | − 1.58·3-s − 0.417·5-s + 0.613·7-s + 1.50·9-s − 1.41·11-s − 0.414·13-s + 0.660·15-s + 0.0750·17-s + 0.730·19-s − 0.971·21-s + 0.925·23-s − 0.825·25-s − 0.802·27-s − 1.66·29-s − 1.47·31-s + 2.23·33-s − 0.256·35-s − 0.503·37-s + 0.656·39-s + 1.90·41-s + 0.684·43-s − 0.628·45-s + 0.382·47-s − 0.623·49-s − 0.118·51-s + 0.749·53-s + 0.589·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 1.61e5T + 1.04e10T^{2} \) |
| 5 | \( 1 + 9.10e6T + 4.76e14T^{2} \) |
| 7 | \( 1 - 4.58e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.21e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.06e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 6.23e11T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.95e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.83e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.78e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 6.71e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.47e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.63e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 9.69e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.37e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 9.54e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 5.91e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.61e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.37e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.70e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.45e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 6.92e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 7.55e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.43e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.23e21T + 5.27e41T^{2} \) |
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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
−10.85311878009381524635319873329, −9.549674131931959648647381942782, −7.84568261688589681192094908986, −7.11091913125867957745737982264, −5.48896313008816240698766918193, −5.25422502482144054469776003842, −3.90069660142459747957908476393, −2.23511253949614482079135176631, −0.849175470625894795157369518695, 0,
0.849175470625894795157369518695, 2.23511253949614482079135176631, 3.90069660142459747957908476393, 5.25422502482144054469776003842, 5.48896313008816240698766918193, 7.11091913125867957745737982264, 7.84568261688589681192094908986, 9.549674131931959648647381942782, 10.85311878009381524635319873329
