L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 7-s − 8-s − 2·9-s − 3·10-s + 12-s + 5·13-s + 14-s + 3·15-s + 16-s + 2·18-s + 2·19-s + 3·20-s − 21-s − 6·23-s − 24-s + 4·25-s − 5·26-s − 5·27-s − 28-s − 3·30-s − 4·31-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 − 2/3·9-s − 0.948·10-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.471·18-s + 0.458·19-s + 0.670·20-s − 0.218·21-s − 1.25·23-s − 0.204·24-s + 4/5·25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s − 0.547·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.123529159\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123529159\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.20723561594024845320951539319, −11.75648260207423678907360760819, −10.63047286146987747081302315269, −9.683357631351870585115325202362, −8.946470375618335270233066681801, −8.008923035003699396981349994067, −6.43698619599801084891178808154, −5.66674862285945396423871921055, −3.35403695021279589879198759951, −1.90147284566448759002651582081,
1.90147284566448759002651582081, 3.35403695021279589879198759951, 5.66674862285945396423871921055, 6.43698619599801084891178808154, 8.008923035003699396981349994067, 8.946470375618335270233066681801, 9.683357631351870585115325202362, 10.63047286146987747081302315269, 11.75648260207423678907360760819, 13.20723561594024845320951539319