L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 2·13-s + 15-s − 6·17-s − 8·19-s − 21-s + 25-s − 27-s − 6·29-s − 4·31-s − 35-s + 10·37-s + 2·39-s − 6·41-s + 4·43-s − 45-s + 49-s + 6·51-s + 6·53-s + 8·57-s + 12·59-s + 10·61-s + 63-s + 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 1.83·19-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 + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 1.28·61-s + 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8320264392\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8320264392\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−8.045677196014288141926184686800, −7.06610334010345228861281720782, −6.74838645501479793062216150111, −5.83436602864038509803013268902, −5.13968965114004266117160383710, −4.24135384027687278383054350642, −3.99543284595006496862105331367, −2.53320519897979202734332076144, −1.90125709661185609560163965495, −0.46302021003691119078923642314,
0.46302021003691119078923642314, 1.90125709661185609560163965495, 2.53320519897979202734332076144, 3.99543284595006496862105331367, 4.24135384027687278383054350642, 5.13968965114004266117160383710, 5.83436602864038509803013268902, 6.74838645501479793062216150111, 7.06610334010345228861281720782, 8.045677196014288141926184686800