L(s) = 1 | − 5·13-s + 19-s − 5·25-s − 11·31-s + 11·37-s − 13·43-s − 14·61-s + 5·67-s − 17·73-s + 17·79-s − 14·97-s + 13·103-s − 19·109-s + ⋯ |
L(s) = 1 | − 1.38·13-s + 0.229·19-s − 25-s − 1.97·31-s + 1.80·37-s − 1.98·43-s − 1.79·61-s + 0.610·67-s − 1.98·73-s + 1.91·79-s − 1.42·97-s + 1.28·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.072445151185707267203043951256, −7.919824911977054215538992032791, −7.46472304636129055466154416467, −6.54001911877925024909239165864, −5.58966265767998501281876191722, −4.83519218317223951659185514643, −3.86532544874840477233066388199, −2.79032016914030089273128181768, −1.74596868449437337311970648766, 0,
1.74596868449437337311970648766, 2.79032016914030089273128181768, 3.86532544874840477233066388199, 4.83519218317223951659185514643, 5.58966265767998501281876191722, 6.54001911877925024909239165864, 7.46472304636129055466154416467, 7.919824911977054215538992032791, 9.072445151185707267203043951256