L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2·11-s − 4·13-s + 15-s − 2·17-s + 4·19-s − 21-s − 23-s + 25-s + 27-s − 4·29-s + 2·33-s − 35-s + 6·37-s − 4·39-s − 10·41-s + 2·43-s + 45-s + 8·47-s + 49-s − 2·51-s − 6·53-s + 2·55-s + 4·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.348·33-s − 0.169·35-s + 0.986·37-s − 0.640·39-s − 1.56·41-s + 0.304·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.269·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.66719320235550, −13.15938972463066, −12.54512280721138, −12.26496349773393, −11.71185608922696, −11.16277888232229, −10.65011843858285, −10.00919334432429, −9.579390405636218, −9.393195392604777, −8.868018946569222, −8.187172917521146, −7.756972052098475, −7.180493624615515, −6.744043978104551, −6.317283469926874, −5.505584016806205, −5.218774819447049, −4.467390900165786, −3.968637323759964, −3.374770205932431, −2.757249622925550, −2.274860653005750, −1.654581091161047, −0.9153250889636469, 0,
0.9153250889636469, 1.654581091161047, 2.274860653005750, 2.757249622925550, 3.374770205932431, 3.968637323759964, 4.467390900165786, 5.218774819447049, 5.505584016806205, 6.317283469926874, 6.744043978104551, 7.180493624615515, 7.756972052098475, 8.187172917521146, 8.868018946569222, 9.393195392604777, 9.579390405636218, 10.00919334432429, 10.65011843858285, 11.16277888232229, 11.71185608922696, 12.26496349773393, 12.54512280721138, 13.15938972463066, 13.66719320235550