| L(s) = 1 | − 0.931·3-s + 0.956·5-s + 3.32·7-s − 2.13·9-s − 1.96·11-s + 2.21·13-s − 0.891·15-s − 6.00·17-s − 4.04·19-s − 3.09·21-s − 4.11·23-s − 4.08·25-s + 4.78·27-s + 10.4·29-s + 7.52·31-s + 1.83·33-s + 3.18·35-s − 2.59·37-s − 2.06·39-s + 1.95·41-s + 11.7·43-s − 2.03·45-s + 3.33·47-s + 4.06·49-s + 5.60·51-s + 8.69·53-s − 1.88·55-s + ⋯ |
| L(s) = 1 | − 0.538·3-s + 0.427·5-s + 1.25·7-s − 0.710·9-s − 0.593·11-s + 0.614·13-s − 0.230·15-s − 1.45·17-s − 0.927·19-s − 0.676·21-s − 0.858·23-s − 0.817·25-s + 0.920·27-s + 1.94·29-s + 1.35·31-s + 0.319·33-s + 0.537·35-s − 0.427·37-s − 0.330·39-s + 0.304·41-s + 1.79·43-s − 0.303·45-s + 0.485·47-s + 0.580·49-s + 0.784·51-s + 1.19·53-s − 0.253·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620753609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620753609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 0.931T + 3T^{2} \) |
| 5 | \( 1 - 0.956T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 + 6.00T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 + 2.59T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.33T + 47T^{2} \) |
| 53 | \( 1 - 8.69T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.179T + 61T^{2} \) |
| 67 | \( 1 + 9.08T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.26T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 - 5.18T + 83T^{2} \) |
| 89 | \( 1 + 9.02T + 89T^{2} \) |
| 97 | \( 1 + 0.874T + 97T^{2} \) |
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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
−8.475663337098403904307846873406, −7.895138638555300011403217349502, −6.78098745072063179252394873413, −6.11859370556648089030240696074, −5.55275516855140888013840993863, −4.64254909095700473655587489524, −4.17435227745639345688756424351, −2.63926173028804236593257362644, −2.06691822145675196820053811706, −0.73746223382731401646624475560,
0.73746223382731401646624475560, 2.06691822145675196820053811706, 2.63926173028804236593257362644, 4.17435227745639345688756424351, 4.64254909095700473655587489524, 5.55275516855140888013840993863, 6.11859370556648089030240696074, 6.78098745072063179252394873413, 7.895138638555300011403217349502, 8.475663337098403904307846873406
