| L(s) = 1 | + 5-s + 7-s + 6·13-s − 2·17-s + 8·19-s − 4·23-s + 25-s + 6·29-s + 4·31-s + 35-s − 2·37-s − 2·41-s + 12·43-s + 49-s − 2·53-s + 4·59-s − 6·61-s + 6·65-s − 4·67-s − 8·71-s − 6·73-s + 16·79-s − 4·83-s − 2·85-s + 18·89-s + 6·91-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 1.82·43-s + 1/7·49-s − 0.274·53-s + 0.520·59-s − 0.768·61-s + 0.744·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s + 1.80·79-s − 0.439·83-s − 0.216·85-s + 1.90·89-s + 0.628·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.588857758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.588857758\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.62417825330028, −12.04910305277411, −11.88502112956148, −11.24915489137420, −10.86832381911353, −10.38720919889787, −10.02120589339974, −9.339756129753913, −9.053545539475780, −8.538681513389040, −8.063764576924764, −7.617293432889393, −7.110345394671678, −6.408117143357007, −6.114062486233130, −5.695701137972401, −5.055333646088013, −4.631980049837234, −3.976742269605177, −3.523297852218326, −2.910761246729892, −2.395930473741585, −1.605687474779605, −1.172701737115260, −0.6211932209128952,
0.6211932209128952, 1.172701737115260, 1.605687474779605, 2.395930473741585, 2.910761246729892, 3.523297852218326, 3.976742269605177, 4.631980049837234, 5.055333646088013, 5.695701137972401, 6.114062486233130, 6.408117143357007, 7.110345394671678, 7.617293432889393, 8.063764576924764, 8.538681513389040, 9.053545539475780, 9.339756129753913, 10.02120589339974, 10.38720919889787, 10.86832381911353, 11.24915489137420, 11.88502112956148, 12.04910305277411, 12.62417825330028
