| L(s) = 1 | + 18.4·2-s + 161.·3-s − 170.·4-s − 1.97e3·5-s + 2.98e3·6-s + 3.62e3·7-s − 1.26e4·8-s + 6.33e3·9-s − 3.65e4·10-s − 3.00e4·11-s − 2.74e4·12-s − 1.88e5·13-s + 6.69e4·14-s − 3.19e5·15-s − 1.46e5·16-s + 6.85e5·17-s + 1.17e5·18-s − 4.40e5·19-s + 3.36e5·20-s + 5.84e5·21-s − 5.55e5·22-s − 9.07e5·23-s − 2.03e6·24-s + 1.96e6·25-s − 3.47e6·26-s − 2.15e6·27-s − 6.16e5·28-s + ⋯ |
| L(s) = 1 | + 0.817·2-s + 1.14·3-s − 0.332·4-s − 1.41·5-s + 0.939·6-s + 0.570·7-s − 1.08·8-s + 0.322·9-s − 1.15·10-s − 0.618·11-s − 0.382·12-s − 1.82·13-s + 0.465·14-s − 1.62·15-s − 0.557·16-s + 1.99·17-s + 0.263·18-s − 0.775·19-s + 0.470·20-s + 0.655·21-s − 0.505·22-s − 0.676·23-s − 1.25·24-s + 1.00·25-s − 1.49·26-s − 0.779·27-s − 0.189·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.87e6T \) |
| good | 2 | \( 1 - 18.4T + 512T^{2} \) |
| 3 | \( 1 - 161.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.97e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.00e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.88e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.85e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.40e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.07e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.92e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 1.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.37e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.18e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.71e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.95e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.03e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.52e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.84e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.19e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.70e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.77e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.96e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.40e6T + 7.60e17T^{2} \) |
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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
−14.13272808996892080166831253511, −12.58340928978353205198007696905, −11.83635144175598299185420284265, −9.854332205567133679401591111118, −8.286254943972420718586793091923, −7.65874952216556441085713313545, −5.12461033327861993494502502033, −3.89890449603083150680251727105, −2.74782461753150060881516511978, 0,
2.74782461753150060881516511978, 3.89890449603083150680251727105, 5.12461033327861993494502502033, 7.65874952216556441085713313545, 8.286254943972420718586793091923, 9.854332205567133679401591111118, 11.83635144175598299185420284265, 12.58340928978353205198007696905, 14.13272808996892080166831253511
