| L(s) = 1 | − 5-s − 4·7-s + 13-s − 17-s + 25-s + 2·29-s + 4·31-s + 4·35-s − 2·37-s + 6·41-s + 4·43-s + 12·47-s + 9·49-s − 6·53-s + 8·59-s + 14·61-s − 65-s + 8·67-s + 12·71-s + 2·73-s − 8·79-s + 85-s − 2·89-s − 4·91-s + 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.277·13-s − 0.242·17-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s + 1.04·59-s + 1.79·61-s − 0.124·65-s + 0.977·67-s + 1.42·71-s + 0.234·73-s − 0.900·79-s + 0.108·85-s − 0.211·89-s − 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657373574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657373574\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.06156923993233, −13.41022747405020, −12.94759184792696, −12.53766729128725, −12.18443977502464, −11.43416874554531, −11.12090666680071, −10.31109285311567, −10.12574316152711, −9.379645573475943, −9.090064752121889, −8.415323924236337, −7.954851927848993, −7.201793651016897, −6.830545007603245, −6.319569143305930, −5.792063325542865, −5.192319117300939, −4.392912035086166, −3.854180005039126, −3.438664147475778, −2.662463163898148, −2.274499514583867, −1.028024618614013, −0.4973386756671419,
0.4973386756671419, 1.028024618614013, 2.274499514583867, 2.662463163898148, 3.438664147475778, 3.854180005039126, 4.392912035086166, 5.192319117300939, 5.792063325542865, 6.319569143305930, 6.830545007603245, 7.201793651016897, 7.954851927848993, 8.415323924236337, 9.090064752121889, 9.379645573475943, 10.12574316152711, 10.31109285311567, 11.12090666680071, 11.43416874554531, 12.18443977502464, 12.53766729128725, 12.94759184792696, 13.41022747405020, 14.06156923993233
