| L(s) = 1 | − 1.55·3-s + 5-s + 3.26·7-s − 0.573·9-s + 3.84·11-s + 4.55·13-s − 1.55·15-s + 3.75·17-s + 4.33·19-s − 5.09·21-s − 6.95·23-s + 25-s + 5.56·27-s + 8.89·29-s + 4.65·31-s − 5.98·33-s + 3.26·35-s + 9.51·37-s − 7.09·39-s − 4.95·41-s + 0.695·43-s − 0.573·45-s − 1.13·47-s + 3.69·49-s − 5.85·51-s + 4.06·53-s + 3.84·55-s + ⋯ |
| L(s) = 1 | − 0.899·3-s + 0.447·5-s + 1.23·7-s − 0.191·9-s + 1.15·11-s + 1.26·13-s − 0.402·15-s + 0.910·17-s + 0.994·19-s − 1.11·21-s − 1.44·23-s + 0.200·25-s + 1.07·27-s + 1.65·29-s + 0.836·31-s − 1.04·33-s + 0.552·35-s + 1.56·37-s − 1.13·39-s − 0.773·41-s + 0.106·43-s − 0.0855·45-s − 0.165·47-s + 0.527·49-s − 0.819·51-s + 0.557·53-s + 0.518·55-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
| \(L(1)\) |
\(\approx\) |
\(2.540792256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.540792256\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.55T + 3T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 - 4.33T + 19T^{2} \) |
| 23 | \( 1 + 6.95T + 23T^{2} \) |
| 29 | \( 1 - 8.89T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 + 4.95T + 41T^{2} \) |
| 43 | \( 1 - 0.695T + 43T^{2} \) |
| 47 | \( 1 + 1.13T + 47T^{2} \) |
| 53 | \( 1 - 4.06T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + 7.64T + 79T^{2} \) |
| 83 | \( 1 + 6.72T + 83T^{2} \) |
| 89 | \( 1 - 4.16T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.044432763585071332987921543225, −6.97085850908849949592257554663, −6.18709335590443487110155560911, −5.89859249494987475703910017096, −5.12010052505496051529294255814, −4.43874870010367395831937161340, −3.64691193007850008350279132492, −2.60985622817071110950436835747, −1.35383190183971138409398932126, −1.01178770995267274335085130518,
1.01178770995267274335085130518, 1.35383190183971138409398932126, 2.60985622817071110950436835747, 3.64691193007850008350279132492, 4.43874870010367395831937161340, 5.12010052505496051529294255814, 5.89859249494987475703910017096, 6.18709335590443487110155560911, 6.97085850908849949592257554663, 8.044432763585071332987921543225
