| L(s) = 1 | − 40.3·2-s + 50.3·3-s + 1.11e3·4-s − 1.13e3·5-s − 2.03e3·6-s + 800.·7-s − 2.44e4·8-s − 1.71e4·9-s + 4.58e4·10-s + 4.68e4·11-s + 5.63e4·12-s + 1.25e5·13-s − 3.23e4·14-s − 5.72e4·15-s + 4.16e5·16-s + 7.50e4·17-s + 6.92e5·18-s + 4.04e5·19-s − 1.27e6·20-s + 4.03e4·21-s − 1.89e6·22-s − 1.49e6·23-s − 1.23e6·24-s − 6.62e5·25-s − 5.06e6·26-s − 1.85e6·27-s + 8.95e5·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.359·3-s + 2.18·4-s − 0.813·5-s − 0.640·6-s + 0.126·7-s − 2.11·8-s − 0.870·9-s + 1.45·10-s + 0.965·11-s + 0.784·12-s + 1.21·13-s − 0.224·14-s − 0.292·15-s + 1.58·16-s + 0.217·17-s + 1.55·18-s + 0.712·19-s − 1.77·20-s + 0.0452·21-s − 1.72·22-s − 1.11·23-s − 0.759·24-s − 0.338·25-s − 2.17·26-s − 0.672·27-s + 0.275·28-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 | \( 1 + 40.3T + 512T^{2} \) |
| 3 | \( 1 - 50.3T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.13e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 800.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.68e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.25e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 7.50e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.49e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.87e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 2.75e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.56e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.04e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.36e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.83e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 7.75e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 7.73e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 9.50e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.71e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.23e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.78e8T + 7.60e17T^{2} \) |
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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
−14.04355015792234910661575562349, −11.81512391305892123334774453327, −11.20904363334268375416427156838, −9.688883045465427953087803794597, −8.558403833552094003845745382527, −7.85468665514006098399709450924, −6.32397588835333919666516891422, −3.43693428072588755204388600184, −1.51859688804889896267310212480, 0,
1.51859688804889896267310212480, 3.43693428072588755204388600184, 6.32397588835333919666516891422, 7.85468665514006098399709450924, 8.558403833552094003845745382527, 9.688883045465427953087803794597, 11.20904363334268375416427156838, 11.81512391305892123334774453327, 14.04355015792234910661575562349
