L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s + 12-s − 4·13-s − 14-s + 3·15-s + 16-s − 3·17-s − 18-s + 2·19-s + 3·20-s + 21-s − 24-s + 4·25-s + 4·26-s + 27-s + 28-s − 3·30-s + 2·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s − 1.10·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.670·20-s + 0.218·21-s − 0.204·24-s + 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s − 0.547·30-s + 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.749975130\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.749975130\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.59335802028594, −14.91221449359705, −14.44537917614712, −13.92734971754096, −13.49968146182566, −12.85636540697363, −12.26973186887528, −11.67458154689847, −10.94562555794797, −10.35892226988752, −9.939518530110818, −9.334289057947968, −9.068326586734631, −8.335998069790671, −7.720395423609793, −7.121958803681305, −6.583876149620801, −5.850781743063366, −5.232469292415361, −4.597746556217680, −3.685924114076517, −2.645388960162625, −2.323580599573636, −1.663746424874478, −0.7209970847480566,
0.7209970847480566, 1.663746424874478, 2.323580599573636, 2.645388960162625, 3.685924114076517, 4.597746556217680, 5.232469292415361, 5.850781743063366, 6.583876149620801, 7.121958803681305, 7.720395423609793, 8.335998069790671, 9.068326586734631, 9.334289057947968, 9.939518530110818, 10.35892226988752, 10.94562555794797, 11.67458154689847, 12.26973186887528, 12.85636540697363, 13.49968146182566, 13.92734971754096, 14.44537917614712, 14.91221449359705, 15.59335802028594