| L(s) = 1 | − 3·3-s − 4·5-s + 6·9-s + 3·11-s − 13-s + 12·15-s + 6·17-s + 4·19-s + 3·23-s + 11·25-s − 9·27-s − 3·29-s − 2·31-s − 9·33-s + 7·37-s + 3·39-s + 9·41-s + 10·43-s − 24·45-s + 6·47-s − 18·51-s + 8·53-s − 12·55-s − 12·57-s + 6·59-s − 13·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s + 2·9-s + 0.904·11-s − 0.277·13-s + 3.09·15-s + 1.45·17-s + 0.917·19-s + 0.625·23-s + 11/5·25-s − 1.73·27-s − 0.557·29-s − 0.359·31-s − 1.56·33-s + 1.15·37-s + 0.480·39-s + 1.40·41-s + 1.52·43-s − 3.57·45-s + 0.875·47-s − 2.52·51-s + 1.09·53-s − 1.61·55-s − 1.58·57-s + 0.781·59-s − 1.66·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.121989686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.121989686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−12.59675749196368, −12.33778961755246, −11.99387227032072, −11.54397417876208, −11.25897490128264, −10.87804296265174, −10.36829968298622, −9.781092256398851, −9.234011536405879, −8.837085060805688, −7.882971155878899, −7.641466396636622, −7.226420473923611, −6.916016656792411, −6.024157768980267, −5.847981307409315, −5.185198859921971, −4.732300585647292, −4.064787209321898, −3.924119236294227, −3.217663629153271, −2.520081908097450, −1.247116269955962, −0.9719413305391692, −0.4622543467068641,
0.4622543467068641, 0.9719413305391692, 1.247116269955962, 2.520081908097450, 3.217663629153271, 3.924119236294227, 4.064787209321898, 4.732300585647292, 5.185198859921971, 5.847981307409315, 6.024157768980267, 6.916016656792411, 7.226420473923611, 7.641466396636622, 7.882971155878899, 8.837085060805688, 9.234011536405879, 9.781092256398851, 10.36829968298622, 10.87804296265174, 11.25897490128264, 11.54397417876208, 11.99387227032072, 12.33778961755246, 12.59675749196368
