| L(s) = 1 | − 3-s − 2·4-s + 3·5-s − 2·9-s + 2·12-s − 3·15-s + 4·16-s − 6·20-s − 9·23-s + 4·25-s + 5·27-s + 5·31-s + 4·36-s − 7·37-s − 6·45-s + 12·47-s − 4·48-s − 7·49-s + 6·53-s + 15·59-s + 6·60-s − 8·64-s − 13·67-s + 9·69-s + 3·71-s − 4·75-s + 12·80-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.34·5-s − 2/3·9-s + 0.577·12-s − 0.774·15-s + 16-s − 1.34·20-s − 1.87·23-s + 4/5·25-s + 0.962·27-s + 0.898·31-s + 2/3·36-s − 1.15·37-s − 0.894·45-s + 1.75·47-s − 0.577·48-s − 49-s + 0.824·53-s + 1.95·59-s + 0.774·60-s − 64-s − 1.58·67-s + 1.08·69-s + 0.356·71-s − 0.461·75-s + 1.34·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| 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
−16.10732998539546, −15.29450753122936, −14.59426551602667, −14.08209299930730, −13.77445041498342, −13.38192266934316, −12.63011300871397, −12.08431855830962, −11.68206723376852, −10.72398152808914, −10.27015892568618, −9.873307867709100, −9.298092837204994, −8.598469995942681, −8.303128853060176, −7.400188387139721, −6.528492307191116, −5.939875254494539, −5.616951453488715, −5.053936534520119, −4.312949550982108, −3.596274921325540, −2.648773186143723, −1.937906770435351, −0.9642076691268300, 0,
0.9642076691268300, 1.937906770435351, 2.648773186143723, 3.596274921325540, 4.312949550982108, 5.053936534520119, 5.616951453488715, 5.939875254494539, 6.528492307191116, 7.400188387139721, 8.303128853060176, 8.598469995942681, 9.298092837204994, 9.873307867709100, 10.27015892568618, 10.72398152808914, 11.68206723376852, 12.08431855830962, 12.63011300871397, 13.38192266934316, 13.77445041498342, 14.08209299930730, 14.59426551602667, 15.29450753122936, 16.10732998539546
