| L(s) = 1 | − 3.20·3-s − 5-s − 2.96·7-s + 7.27·9-s + 0.803·11-s − 0.822·13-s + 3.20·15-s − 4.32·17-s − 5.88·19-s + 9.50·21-s − 3.95·23-s + 25-s − 13.7·27-s + 8.32·29-s − 4.02·31-s − 2.57·33-s + 2.96·35-s + 2.20·37-s + 2.63·39-s − 0.284·41-s + 3.27·43-s − 7.27·45-s − 0.459·47-s + 1.78·49-s + 13.8·51-s − 12.2·53-s − 0.803·55-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 0.447·5-s − 1.12·7-s + 2.42·9-s + 0.242·11-s − 0.228·13-s + 0.827·15-s − 1.04·17-s − 1.34·19-s + 2.07·21-s − 0.823·23-s + 0.200·25-s − 2.63·27-s + 1.54·29-s − 0.723·31-s − 0.448·33-s + 0.501·35-s + 0.362·37-s + 0.422·39-s − 0.0443·41-s + 0.499·43-s − 1.08·45-s − 0.0670·47-s + 0.255·49-s + 1.94·51-s − 1.67·53-s − 0.108·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1007421499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1007421499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 3.20T + 3T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 - 0.803T + 11T^{2} \) |
| 13 | \( 1 + 0.822T + 13T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 37 | \( 1 - 2.20T + 37T^{2} \) |
| 41 | \( 1 + 0.284T + 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 + 0.459T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 0.769T + 61T^{2} \) |
| 67 | \( 1 + 5.42T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 + 1.80T + 73T^{2} \) |
| 79 | \( 1 - 1.11T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57682372341659672855529191088, −6.81513192219626498247742066613, −6.30383549574840575456797600737, −6.09112941048815180206573288584, −4.98535803359656113530663080037, −4.42155698269647865456951366187, −3.84243615590588230756601213885, −2.65313309942657447622686297769, −1.46652411083440278197791835959, −0.17701834072346213945570181838,
0.17701834072346213945570181838, 1.46652411083440278197791835959, 2.65313309942657447622686297769, 3.84243615590588230756601213885, 4.42155698269647865456951366187, 4.98535803359656113530663080037, 6.09112941048815180206573288584, 6.30383549574840575456797600737, 6.81513192219626498247742066613, 7.57682372341659672855529191088
