| L(s) = 1 | − 3·7-s − 5·11-s − 4·13-s − 8·17-s + 2·19-s + 2·23-s − 6·29-s − 7·31-s + 6·37-s + 6·41-s + 2·43-s + 6·47-s + 2·49-s + 5·53-s + 4·59-s − 8·61-s + 10·67-s + 8·71-s − 73-s + 15·77-s + 16·79-s − 11·83-s − 6·89-s + 12·91-s + 97-s − 9·101-s − 4·103-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1.50·11-s − 1.10·13-s − 1.94·17-s + 0.458·19-s + 0.417·23-s − 1.11·29-s − 1.25·31-s + 0.986·37-s + 0.937·41-s + 0.304·43-s + 0.875·47-s + 2/7·49-s + 0.686·53-s + 0.520·59-s − 1.02·61-s + 1.22·67-s + 0.949·71-s − 0.117·73-s + 1.70·77-s + 1.80·79-s − 1.20·83-s − 0.635·89-s + 1.25·91-s + 0.101·97-s − 0.895·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6212903381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6212903381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.069269421184808740710970697688, −7.33268776082441586263181416431, −6.92125321323708344200732959335, −5.96594090077212321671272338363, −5.31975168575974286863951852386, −4.54769035050128794925612182778, −3.65487943004336719014507479967, −2.63349223909005619441201203248, −2.24627128455249706881676401018, −0.38851179148025330850270876693,
0.38851179148025330850270876693, 2.24627128455249706881676401018, 2.63349223909005619441201203248, 3.65487943004336719014507479967, 4.54769035050128794925612182778, 5.31975168575974286863951852386, 5.96594090077212321671272338363, 6.92125321323708344200732959335, 7.33268776082441586263181416431, 8.069269421184808740710970697688
