| L(s) = 1 | + 2·3-s − 5-s + 4·7-s + 9-s + 2·11-s − 2·13-s − 2·15-s + 4·19-s + 8·21-s − 4·23-s + 25-s − 4·27-s − 6·29-s + 4·33-s − 4·35-s − 10·37-s − 4·39-s − 10·41-s − 6·43-s − 45-s + 4·47-s + 9·49-s + 6·53-s − 2·55-s + 8·57-s − 12·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.516·15-s + 0.917·19-s + 1.74·21-s − 0.834·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.696·33-s − 0.676·35-s − 1.64·37-s − 0.640·39-s − 1.56·41-s − 0.914·43-s − 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.269·55-s + 1.05·57-s − 1.56·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.02409656855102, −14.49429041780105, −14.22626196372296, −13.61371676406462, −13.34845745094895, −12.24166207486680, −11.97222183371271, −11.55106292339234, −10.95785205511965, −10.28834488234908, −9.711164157244345, −9.058707545286704, −8.654726690351911, −8.127963168137723, −7.701852186648505, −7.246788072106242, −6.598398865802095, −5.549742718249363, −5.157562048439374, −4.505793256702240, −3.639840052527969, −3.511758479687734, −2.434868572230903, −1.888834111055826, −1.302675762678430, 0,
1.302675762678430, 1.888834111055826, 2.434868572230903, 3.511758479687734, 3.639840052527969, 4.505793256702240, 5.157562048439374, 5.549742718249363, 6.598398865802095, 7.246788072106242, 7.701852186648505, 8.127963168137723, 8.654726690351911, 9.058707545286704, 9.711164157244345, 10.28834488234908, 10.95785205511965, 11.55106292339234, 11.97222183371271, 12.24166207486680, 13.34845745094895, 13.61371676406462, 14.22626196372296, 14.49429041780105, 15.02409656855102
