L(s) = 1 | + 5-s − 7-s − 2·11-s − 4·13-s − 3·17-s + 2·19-s + 6·23-s − 4·25-s − 6·29-s − 4·31-s − 35-s − 11·37-s − 5·41-s − 7·43-s + 5·47-s + 49-s + 6·53-s − 2·55-s + 9·59-s + 6·61-s − 4·65-s − 4·67-s + 8·71-s − 2·73-s + 2·77-s − 11·79-s − 5·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.603·11-s − 1.10·13-s − 0.727·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s − 1.11·29-s − 0.718·31-s − 0.169·35-s − 1.80·37-s − 0.780·41-s − 1.06·43-s + 0.729·47-s + 1/7·49-s + 0.824·53-s − 0.269·55-s + 1.17·59-s + 0.768·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 0.234·73-s + 0.227·77-s − 1.23·79-s − 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.173063642519491149099969961659, −8.362913963475538918276135488476, −7.22077263548401962038411801590, −6.88967708000099328016942093759, −5.53738241791908946234781007711, −5.15557583800669337801512234558, −3.88891052691857294431929206040, −2.82146121048544761245626442593, −1.85632335524353518843856378045, 0,
1.85632335524353518843856378045, 2.82146121048544761245626442593, 3.88891052691857294431929206040, 5.15557583800669337801512234558, 5.53738241791908946234781007711, 6.88967708000099328016942093759, 7.22077263548401962038411801590, 8.362913963475538918276135488476, 9.173063642519491149099969961659