| L(s) = 1 | − 5-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s + 8·23-s + 25-s − 2·29-s − 6·37-s − 6·41-s + 4·43-s − 10·53-s + 4·55-s − 12·59-s + 14·61-s + 2·65-s + 12·67-s + 8·71-s − 10·73-s + 16·79-s + 12·83-s − 2·85-s + 10·89-s − 4·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 1.37·53-s + 0.539·55-s − 1.56·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 1.31·83-s − 0.216·85-s + 1.05·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.66898620282332, −13.07880382936120, −12.67528192827817, −12.27588109630665, −11.75615205509227, −11.13859348621781, −10.89435029711502, −10.22978512500784, −9.904973555637528, −9.173639834895507, −8.937970786762675, −8.117728091916702, −7.695541346179940, −7.488998003630990, −6.736435960546261, −6.384174194115112, −5.355221061102027, −5.120804928374248, −4.919340053073840, −3.876244571472878, −3.437548567188541, −2.863133089219199, −2.368872469223283, −1.497828030852549, −0.7711135888447305, 0,
0.7711135888447305, 1.497828030852549, 2.368872469223283, 2.863133089219199, 3.437548567188541, 3.876244571472878, 4.919340053073840, 5.120804928374248, 5.355221061102027, 6.384174194115112, 6.736435960546261, 7.488998003630990, 7.695541346179940, 8.117728091916702, 8.937970786762675, 9.173639834895507, 9.904973555637528, 10.22978512500784, 10.89435029711502, 11.13859348621781, 11.75615205509227, 12.27588109630665, 12.67528192827817, 13.07880382936120, 13.66898620282332
