| L(s) = 1 | + 19.7·2-s + 250.·3-s − 123.·4-s + 1.55e3·5-s + 4.93e3·6-s − 8.32e3·7-s − 1.25e4·8-s + 4.30e4·9-s + 3.06e4·10-s + 3.07e4·11-s − 3.08e4·12-s + 2.85e4·13-s − 1.64e5·14-s + 3.89e5·15-s − 1.83e5·16-s − 6.37e5·17-s + 8.48e5·18-s + 1.05e5·19-s − 1.91e5·20-s − 2.08e6·21-s + 6.06e5·22-s − 5.11e5·23-s − 3.13e6·24-s + 4.66e5·25-s + 5.63e5·26-s + 5.85e6·27-s + 1.02e6·28-s + ⋯ |
| L(s) = 1 | + 0.871·2-s + 1.78·3-s − 0.240·4-s + 1.11·5-s + 1.55·6-s − 1.31·7-s − 1.08·8-s + 2.18·9-s + 0.969·10-s + 0.633·11-s − 0.429·12-s + 0.277·13-s − 1.14·14-s + 1.98·15-s − 0.701·16-s − 1.85·17-s + 1.90·18-s + 0.186·19-s − 0.267·20-s − 2.34·21-s + 0.551·22-s − 0.380·23-s − 1.93·24-s + 0.238·25-s + 0.241·26-s + 2.12·27-s + 0.315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.703523728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.703523728\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 19.7T + 512T^{2} \) |
| 3 | \( 1 - 250.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.32e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.07e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 6.37e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.11e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.81e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.83e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.83e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.84e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.30e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.63e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.90e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.22e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.65e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.42e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.64e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.98e7T + 7.60e17T^{2} \) |
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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
−17.94519099041213203106896708155, −15.74541872677088976201740041247, −14.45482007432115288086510408978, −13.49504937934845821201429638115, −12.95942396114845730140325352163, −9.640976171901390060997929183005, −8.955033076950461752009246256058, −6.43043112468585301663273465919, −3.92162465517986363555414252976, −2.48308230655511526929776622450,
2.48308230655511526929776622450, 3.92162465517986363555414252976, 6.43043112468585301663273465919, 8.955033076950461752009246256058, 9.640976171901390060997929183005, 12.95942396114845730140325352163, 13.49504937934845821201429638115, 14.45482007432115288086510408978, 15.74541872677088976201740041247, 17.94519099041213203106896708155
