| L(s) = 1 | − 5-s + 4·11-s + 4·13-s + 4·17-s + 4·19-s + 23-s + 25-s + 6·29-s + 4·31-s − 2·37-s − 4·43-s − 7·49-s + 6·53-s − 4·55-s + 6·59-s − 4·61-s − 4·65-s + 4·67-s + 4·71-s − 2·73-s + 10·79-s − 18·83-s − 4·85-s + 2·89-s − 4·95-s + 2·97-s − 10·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s + 1.10·13-s + 0.970·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.609·43-s − 49-s + 0.824·53-s − 0.539·55-s + 0.781·59-s − 0.512·61-s − 0.496·65-s + 0.488·67-s + 0.474·71-s − 0.234·73-s + 1.12·79-s − 1.97·83-s − 0.433·85-s + 0.211·89-s − 0.410·95-s + 0.203·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.527275929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527275929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.952622053400345895853336555928, −6.98211914064101630116983992196, −6.55351916630186942097714345747, −5.76450015957765016051124357401, −5.01750292973542501929516188251, −4.13439793919408799983646985271, −3.53671040349703100180349980856, −2.87762619909490593213783942756, −1.48772127093304428738896342853, −0.884879612853482062443954632893,
0.884879612853482062443954632893, 1.48772127093304428738896342853, 2.87762619909490593213783942756, 3.53671040349703100180349980856, 4.13439793919408799983646985271, 5.01750292973542501929516188251, 5.76450015957765016051124357401, 6.55351916630186942097714345747, 6.98211914064101630116983992196, 7.952622053400345895853336555928
