L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 11-s + 4·13-s − 15-s + 4·19-s + 2·21-s − 4·23-s + 25-s + 27-s − 2·29-s + 8·31-s − 33-s − 2·35-s − 2·37-s + 4·39-s − 6·41-s + 6·43-s − 45-s + 4·47-s − 3·49-s + 6·53-s + 55-s + 4·57-s + 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.258·15-s + 0.917·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.174·33-s − 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.914·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.824·53-s + 0.134·55-s + 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.435416262\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.435416262\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.662553409330316411352008572618, −8.124200373916318946876915545517, −7.59364783098817535036426542022, −6.66604855731502455191889780012, −5.73056516914782841205592730177, −4.84197780746655664894845120544, −3.98475908501671696465668221497, −3.23600988920277730431396057221, −2.12435257245963069973622320134, −1.00992998288259331163718512962,
1.00992998288259331163718512962, 2.12435257245963069973622320134, 3.23600988920277730431396057221, 3.98475908501671696465668221497, 4.84197780746655664894845120544, 5.73056516914782841205592730177, 6.66604855731502455191889780012, 7.59364783098817535036426542022, 8.124200373916318946876915545517, 8.662553409330316411352008572618