| L(s) = 1 | − 1.75e6·3-s − 1.37e8·5-s + 3.04e10·7-s + 2.24e12·9-s − 2.58e12·11-s − 9.57e13·13-s + 2.40e14·15-s − 1.64e15·17-s − 4.95e15·19-s − 5.34e16·21-s + 1.07e16·23-s − 2.79e17·25-s − 2.45e18·27-s − 1.36e18·29-s + 4.41e18·31-s + 4.54e18·33-s − 4.16e18·35-s + 1.01e19·37-s + 1.68e20·39-s + 1.58e20·41-s − 1.83e20·43-s − 3.07e20·45-s − 1.40e21·47-s − 4.15e20·49-s + 2.89e21·51-s − 1.99e21·53-s + 3.54e20·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s − 0.251·5-s + 0.830·7-s + 2.64·9-s − 0.248·11-s − 1.13·13-s + 0.479·15-s − 0.685·17-s − 0.513·19-s − 1.58·21-s + 0.102·23-s − 0.936·25-s − 3.14·27-s − 0.717·29-s + 1.00·31-s + 0.474·33-s − 0.208·35-s + 0.254·37-s + 2.17·39-s + 1.09·41-s − 0.700·43-s − 0.664·45-s − 1.76·47-s − 0.310·49-s + 1.30·51-s − 0.558·53-s + 0.0623·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.5901131402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5901131402\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + 1.75e6T + 8.47e11T^{2} \) |
| 5 | \( 1 + 1.37e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 3.04e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 2.58e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 9.57e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.64e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 4.95e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.07e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.36e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 4.41e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.01e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.58e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.83e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.40e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 1.99e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 4.16e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.42e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 8.67e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 5.13e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.49e22T + 3.82e46T^{2} \) |
| 79 | \( 1 + 2.91e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.64e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 8.74e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.00e25T + 4.66e49T^{2} \) |
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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
−13.00108795312898002748002726416, −11.83373950796997461601512574218, −11.10728760474328901753007745851, −9.906756360376971332201207381686, −7.72919487931930093605223924110, −6.47674008798368008279193010116, −5.18097272429796039104452829655, −4.37794473092975835173861452757, −1.87338149733849407912840329629, −0.44782006828610465679020928063,
0.44782006828610465679020928063, 1.87338149733849407912840329629, 4.37794473092975835173861452757, 5.18097272429796039104452829655, 6.47674008798368008279193010116, 7.72919487931930093605223924110, 9.906756360376971332201207381686, 11.10728760474328901753007745851, 11.83373950796997461601512574218, 13.00108795312898002748002726416
