| L(s) = 1 | + 2-s + 4-s − 5-s − 3·7-s + 8-s − 10-s − 13-s − 3·14-s + 16-s − 3·17-s + 4·19-s − 20-s + 8·23-s + 25-s − 26-s − 3·28-s + 9·29-s − 4·31-s + 32-s − 3·34-s + 3·35-s − 2·37-s + 4·38-s − 40-s − 10·41-s + 11·43-s + 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.801·14-s + 1/4·16-s − 0.727·17-s + 0.917·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s − 0.196·26-s − 0.566·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.507·35-s − 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.56·41-s + 1.67·43-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.515718498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.515718498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−13.39305749174529, −12.90562930670615, −12.52342623619220, −12.01240197431221, −11.69370859883360, −10.95631369477344, −10.67353360056494, −10.13715407299369, −9.463267126972963, −9.112980643829802, −8.625506395887154, −7.877345133434991, −7.314238002573500, −7.019023605284911, −6.343363196871831, −6.148071016327690, −5.270036812687205, −4.767891805091567, −4.491400363408889, −3.537373257626389, −3.186615310716629, −2.883528524678222, −2.040342229378243, −1.204420303710941, −0.4357960398624866,
0.4357960398624866, 1.204420303710941, 2.040342229378243, 2.883528524678222, 3.186615310716629, 3.537373257626389, 4.491400363408889, 4.767891805091567, 5.270036812687205, 6.148071016327690, 6.343363196871831, 7.019023605284911, 7.314238002573500, 7.877345133434991, 8.625506395887154, 9.112980643829802, 9.463267126972963, 10.13715407299369, 10.67353360056494, 10.95631369477344, 11.69370859883360, 12.01240197431221, 12.52342623619220, 12.90562930670615, 13.39305749174529
