| L(s) = 1 | − 5-s + 2·11-s + 13-s + 4·17-s − 19-s − 4·23-s + 25-s − 5·31-s − 5·37-s − 2·41-s − 9·43-s + 2·47-s − 12·53-s − 2·55-s + 8·59-s − 14·61-s − 65-s + 9·67-s − 2·71-s + 73-s − 3·79-s + 18·83-s − 4·85-s + 4·89-s + 95-s + 10·97-s − 6·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s + 0.277·13-s + 0.970·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.898·31-s − 0.821·37-s − 0.312·41-s − 1.37·43-s + 0.291·47-s − 1.64·53-s − 0.269·55-s + 1.04·59-s − 1.79·61-s − 0.124·65-s + 1.09·67-s − 0.237·71-s + 0.117·73-s − 0.337·79-s + 1.97·83-s − 0.433·85-s + 0.423·89-s + 0.102·95-s + 1.01·97-s − 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.51441444199240092505867603473, −6.68723194138330302055960411300, −6.13107958140764687931896593345, −5.30600412310988084273065070739, −4.61699613247306940342757957150, −3.65457817257208641149655015126, −3.36711943082168207493243156146, −2.10032184305076605234204631432, −1.26446692098214953098649762370, 0,
1.26446692098214953098649762370, 2.10032184305076605234204631432, 3.36711943082168207493243156146, 3.65457817257208641149655015126, 4.61699613247306940342757957150, 5.30600412310988084273065070739, 6.13107958140764687931896593345, 6.68723194138330302055960411300, 7.51441444199240092505867603473
