| L(s) = 1 | + 1.51·2-s + 2.59·3-s + 0.284·4-s − 2.07·5-s + 3.91·6-s − 3.84·7-s − 2.59·8-s + 3.72·9-s − 3.13·10-s + 1.14·11-s + 0.737·12-s − 5.25·13-s − 5.81·14-s − 5.37·15-s − 4.48·16-s + 2.41·17-s + 5.62·18-s + 2.85·19-s − 0.590·20-s − 9.97·21-s + 1.72·22-s − 3.59·23-s − 6.72·24-s − 0.694·25-s − 7.94·26-s + 1.87·27-s − 1.09·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1.49·3-s + 0.142·4-s − 0.927·5-s + 1.59·6-s − 1.45·7-s − 0.916·8-s + 1.24·9-s − 0.991·10-s + 0.343·11-s + 0.213·12-s − 1.45·13-s − 1.55·14-s − 1.38·15-s − 1.12·16-s + 0.585·17-s + 1.32·18-s + 0.654·19-s − 0.132·20-s − 2.17·21-s + 0.367·22-s − 0.750·23-s − 1.37·24-s − 0.138·25-s − 1.55·26-s + 0.360·27-s − 0.207·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.51T + 2T^{2} \) |
| 3 | \( 1 - 2.59T + 3T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 + 5.25T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 9.93T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 47 | \( 1 + 0.413T + 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 + 9.98T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 0.792T + 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 - 0.475T + 83T^{2} \) |
| 89 | \( 1 - 4.99T + 89T^{2} \) |
| 97 | \( 1 + 0.449T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.011513319036193080375058976697, −7.86561397097472124551450250578, −7.44802015078167584229394799371, −6.46505918540777509665789494743, −5.47077590766514915571295483710, −4.35840797989242857393314220050, −3.61599129817796298328831788669, −3.23050180744522602409351994913, −2.33699833280299867844139705346, 0,
2.33699833280299867844139705346, 3.23050180744522602409351994913, 3.61599129817796298328831788669, 4.35840797989242857393314220050, 5.47077590766514915571295483710, 6.46505918540777509665789494743, 7.44802015078167584229394799371, 7.86561397097472124551450250578, 9.011513319036193080375058976697
