| L(s) = 1 | − 3·3-s − 5·5-s − 28·7-s + 9·9-s − 24·11-s − 70·13-s + 15·15-s + 102·17-s + 20·19-s + 84·21-s − 72·23-s + 25·25-s − 27·27-s + 306·29-s − 136·31-s + 72·33-s + 140·35-s − 214·37-s + 210·39-s − 150·41-s − 292·43-s − 45·45-s − 72·47-s + 441·49-s − 306·51-s − 414·53-s + 120·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.657·11-s − 1.49·13-s + 0.258·15-s + 1.45·17-s + 0.241·19-s + 0.872·21-s − 0.652·23-s + 1/5·25-s − 0.192·27-s + 1.95·29-s − 0.787·31-s + 0.379·33-s + 0.676·35-s − 0.950·37-s + 0.862·39-s − 0.571·41-s − 1.03·43-s − 0.149·45-s − 0.223·47-s + 9/7·49-s − 0.840·51-s − 1.07·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 306 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 744 T + p^{3} T^{2} \) |
| 61 | \( 1 + 418 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 - 480 T + p^{3} T^{2} \) |
| 73 | \( 1 - 434 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
| 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
−13.97883824605618506336793711989, −12.52264828827559888264095313060, −12.12621169093879686834099842228, −10.37157843884085458609714185865, −9.668125375965989003262875060036, −7.78806732533147135536804407626, −6.58165077951551558758762791573, −5.10950231974509582749830592934, −3.18588889555411344031624098787, 0,
3.18588889555411344031624098787, 5.10950231974509582749830592934, 6.58165077951551558758762791573, 7.78806732533147135536804407626, 9.668125375965989003262875060036, 10.37157843884085458609714185865, 12.12621169093879686834099842228, 12.52264828827559888264095313060, 13.97883824605618506336793711989
