| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 6·11-s − 12-s − 13-s − 14-s + 16-s + 17-s + 18-s − 6·19-s + 21-s + 6·22-s − 4·23-s − 24-s − 26-s − 27-s − 28-s + 4·29-s − 4·31-s + 32-s − 6·33-s + 34-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 + 1.80·11-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 − 1.37·19-s + 0.218·21-s + 1.27·22-s − 0.834·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s − 1.04·33-s + 0.171·34-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.09036239104660, −12.51503268320737, −12.23611009281631, −11.91203828362150, −11.48571567025650, −10.84777571999092, −10.47001340722299, −10.05764337202630, −9.418526569033141, −8.998672942775841, −8.489167422752406, −7.920323178624142, −7.155369995106719, −6.831484924033622, −6.423460813578107, −6.051309641606902, −5.454292180837073, −4.965096954778533, −4.236596038949827, −3.975370328713279, −3.548374582176091, −2.785805872867200, −2.016581887145315, −1.624595137183882, −0.8339765139779465, 0,
0.8339765139779465, 1.624595137183882, 2.016581887145315, 2.785805872867200, 3.548374582176091, 3.975370328713279, 4.236596038949827, 4.965096954778533, 5.454292180837073, 6.051309641606902, 6.423460813578107, 6.831484924033622, 7.155369995106719, 7.920323178624142, 8.489167422752406, 8.998672942775841, 9.418526569033141, 10.05764337202630, 10.47001340722299, 10.84777571999092, 11.48571567025650, 11.91203828362150, 12.23611009281631, 12.51503268320737, 13.09036239104660
