L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 15-s + 6·17-s − 4·19-s − 21-s + 25-s − 27-s + 6·29-s − 4·31-s − 35-s + 10·37-s − 6·41-s − 8·43-s − 45-s + 49-s − 6·51-s − 6·53-s + 4·57-s − 12·59-s + 14·61-s + 63-s − 4·67-s − 2·73-s − 75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.169·35-s + 1.64·37-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s + 0.125·63-s − 0.488·67-s − 0.234·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.90772005208802, −12.42720620803060, −12.01299692027884, −11.59228554293219, −11.22426870272326, −10.70908999700293, −10.22328532295311, −9.891272458434288, −9.365963909161193, −8.660632224336367, −8.280615966269825, −7.863551899172894, −7.404519573411846, −6.801437381816584, −6.413990823529794, −5.819290064991061, −5.386207955382366, −4.841518752894644, −4.359240072518913, −3.926408861242010, −3.166271180293409, −2.833077371706706, −1.892626107984032, −1.406801732114246, −0.7381882169339960, 0,
0.7381882169339960, 1.406801732114246, 1.892626107984032, 2.833077371706706, 3.166271180293409, 3.926408861242010, 4.359240072518913, 4.841518752894644, 5.386207955382366, 5.819290064991061, 6.413990823529794, 6.801437381816584, 7.404519573411846, 7.863551899172894, 8.280615966269825, 8.660632224336367, 9.365963909161193, 9.891272458434288, 10.22328532295311, 10.70908999700293, 11.22426870272326, 11.59228554293219, 12.01299692027884, 12.42720620803060, 12.90772005208802