| L(s) = 1 | − 3-s − 5-s + 9-s − 2·13-s + 15-s + 2·17-s − 4·19-s + 8·23-s + 25-s − 27-s − 2·29-s − 4·31-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s − 2·51-s + 6·53-s + 4·57-s − 6·61-s + 2·65-s + 4·67-s − 8·69-s + 8·71-s − 10·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.768·61-s + 0.248·65-s + 0.488·67-s − 0.963·69-s + 0.949·71-s − 1.17·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.62332951501682, −15.23167994366870, −14.69840477388342, −14.27344784891751, −13.37379083893482, −12.96344694816166, −12.40344544628448, −12.00066708794425, −11.29470635261650, −10.86684302407198, −10.40098214138132, −9.707885488243425, −9.047151642666441, −8.602511776401740, −7.745586660408248, −7.303914853513333, −6.742555368732883, −6.117840972650302, −5.271761625870936, −4.972466982231702, −4.149616191790288, −3.527353687640934, −2.737325070926717, −1.865584372428594, −0.9234994320078160, 0,
0.9234994320078160, 1.865584372428594, 2.737325070926717, 3.527353687640934, 4.149616191790288, 4.972466982231702, 5.271761625870936, 6.117840972650302, 6.742555368732883, 7.303914853513333, 7.745586660408248, 8.602511776401740, 9.047151642666441, 9.707885488243425, 10.40098214138132, 10.86684302407198, 11.29470635261650, 12.00066708794425, 12.40344544628448, 12.96344694816166, 13.37379083893482, 14.27344784891751, 14.69840477388342, 15.23167994366870, 15.62332951501682
