L(s) = 1 | − 2·7-s − 2·13-s − 6·17-s − 4·19-s + 6·23-s − 6·29-s − 4·31-s − 2·37-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s − 6·53-s − 12·59-s + 2·61-s − 2·67-s + 12·71-s − 2·73-s + 8·79-s + 6·83-s + 6·89-s + 4·91-s − 2·97-s − 6·101-s − 14·103-s − 6·107-s + 2·109-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s − 0.244·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s + 0.658·83-s + 0.635·89-s + 0.419·91-s − 0.203·97-s − 0.597·101-s − 1.37·103-s − 0.580·107-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.447689111417190951947467353722, −9.113263281253576510886575085659, −8.008294857926527437201173134092, −6.96167091691043492420755836355, −6.41842389830764202453884824049, −5.24854435453179347418331807133, −4.28530365432405873873397666950, −3.16807967427295861033734664069, −2.01836233216080453835298575952, 0,
2.01836233216080453835298575952, 3.16807967427295861033734664069, 4.28530365432405873873397666950, 5.24854435453179347418331807133, 6.41842389830764202453884824049, 6.96167091691043492420755836355, 8.008294857926527437201173134092, 9.113263281253576510886575085659, 9.447689111417190951947467353722