| L(s) = 1 | − 5-s − 11-s − 2·17-s − 6·19-s + 8·23-s + 25-s + 4·31-s − 2·37-s + 8·41-s − 8·43-s + 12·47-s + 10·53-s + 55-s + 12·59-s + 2·61-s + 4·67-s − 8·71-s + 12·73-s + 10·79-s + 2·83-s + 2·85-s − 6·89-s + 6·95-s + 18·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.485·17-s − 1.37·19-s + 1.66·23-s + 1/5·25-s + 0.718·31-s − 0.328·37-s + 1.24·41-s − 1.21·43-s + 1.75·47-s + 1.37·53-s + 0.134·55-s + 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.949·71-s + 1.40·73-s + 1.12·79-s + 0.219·83-s + 0.216·85-s − 0.635·89-s + 0.615·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.709008839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709008839\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.51814067098267, −11.97349528076205, −11.54233790595877, −11.08742901107645, −10.66437208175026, −10.35036269163265, −9.801446810898256, −9.161152730304030, −8.714251810505912, −8.531685613544950, −7.931836589258297, −7.294496758422237, −7.061373777853685, −6.469518460522211, −6.085263698511262, −5.368134898098398, −4.961820143631641, −4.489445982923886, −3.903509124413023, −3.562945710582975, −2.688109863989043, −2.457572459762353, −1.795372957842447, −0.8390798483790890, −0.5484749194564710,
0.5484749194564710, 0.8390798483790890, 1.795372957842447, 2.457572459762353, 2.688109863989043, 3.562945710582975, 3.903509124413023, 4.489445982923886, 4.961820143631641, 5.368134898098398, 6.085263698511262, 6.469518460522211, 7.061373777853685, 7.294496758422237, 7.931836589258297, 8.531685613544950, 8.714251810505912, 9.161152730304030, 9.801446810898256, 10.35036269163265, 10.66437208175026, 11.08742901107645, 11.54233790595877, 11.97349528076205, 12.51814067098267
