L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 4·11-s − 12-s − 13-s − 14-s + 16-s − 17-s − 18-s + 4·19-s − 21-s + 4·22-s − 8·23-s + 24-s + 26-s − 27-s + 28-s − 2·29-s − 32-s + 4·33-s + 34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.176·32-s + 0.696·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6268132378\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6268132378\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.82345679674187, −12.32683686361966, −11.87313012179426, −11.55062099689255, −10.96564664426636, −10.56195754416871, −10.11923820231872, −9.874403275265632, −9.035715707031412, −8.893629562815060, −8.008791666523778, −7.718162457232222, −7.455757369371641, −6.829146233318977, −6.173886672835227, −5.691496470541413, −5.379429700050559, −4.749242053661936, −4.170562714264310, −3.585373071271735, −2.791248579321449, −2.312424738524693, −1.758970218540645, −1.010284616605529, −0.2860175941342883,
0.2860175941342883, 1.010284616605529, 1.758970218540645, 2.312424738524693, 2.791248579321449, 3.585373071271735, 4.170562714264310, 4.749242053661936, 5.379429700050559, 5.691496470541413, 6.173886672835227, 6.829146233318977, 7.455757369371641, 7.718162457232222, 8.008791666523778, 8.893629562815060, 9.035715707031412, 9.874403275265632, 10.11923820231872, 10.56195754416871, 10.96564664426636, 11.55062099689255, 11.87313012179426, 12.32683686361966, 12.82345679674187