L(s) = 1 | − 4·7-s − 4·11-s − 4·17-s + 4·23-s + 6·29-s − 4·31-s − 8·37-s + 10·41-s − 4·43-s − 4·47-s + 9·49-s + 12·53-s + 4·59-s + 2·61-s + 4·67-s − 8·73-s + 16·77-s + 12·79-s + 4·83-s + 10·89-s + 8·97-s + 2·101-s + 4·103-s + 12·107-s − 2·109-s − 12·113-s + 16·119-s + ⋯ |
L(s) = 1 | − 1.51·7-s − 1.20·11-s − 0.970·17-s + 0.834·23-s + 1.11·29-s − 0.718·31-s − 1.31·37-s + 1.56·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s + 1.64·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.936·73-s + 1.82·77-s + 1.35·79-s + 0.439·83-s + 1.05·89-s + 0.812·97-s + 0.199·101-s + 0.394·103-s + 1.16·107-s − 0.191·109-s − 1.12·113-s + 1.46·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9829520606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9829520606\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.722279788902260758666268829352, −7.74436937528669404580860283143, −6.95350985547462503217842088622, −6.45273776457601220315266125660, −5.55591635467014692996275165189, −4.82431881473241113523540102210, −3.76494698631811766379354328238, −2.97457464677723270922877169003, −2.24554025192167961234838442903, −0.54794063988123240269750231164,
0.54794063988123240269750231164, 2.24554025192167961234838442903, 2.97457464677723270922877169003, 3.76494698631811766379354328238, 4.82431881473241113523540102210, 5.55591635467014692996275165189, 6.45273776457601220315266125660, 6.95350985547462503217842088622, 7.74436937528669404580860283143, 8.722279788902260758666268829352