| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 3·7-s − 2·10-s − 11-s − 2·13-s + 6·14-s − 4·16-s − 2·17-s + 6·19-s + 2·20-s + 2·22-s − 3·23-s − 4·25-s + 4·26-s − 6·28-s − 3·29-s + 2·31-s + 8·32-s + 4·34-s − 3·35-s − 7·37-s − 12·38-s − 10·41-s − 10·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 1.13·7-s − 0.632·10-s − 0.301·11-s − 0.554·13-s + 1.60·14-s − 16-s − 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.426·22-s − 0.625·23-s − 4/5·25-s + 0.784·26-s − 1.13·28-s − 0.557·29-s + 0.359·31-s + 1.41·32-s + 0.685·34-s − 0.507·35-s − 1.15·37-s − 1.94·38-s − 1.56·41-s − 1.52·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 423 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 423 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−10.21139906970486979508175306560, −9.827535245480289108967102730184, −9.149156412188436581192825917436, −8.100801554629335905381784580980, −7.19598013314039086114174891084, −6.33758757673418292336292093348, −5.05201755691324140880779008098, −3.33358434652105144081845579676, −1.89809557201657584670009427079, 0,
1.89809557201657584670009427079, 3.33358434652105144081845579676, 5.05201755691324140880779008098, 6.33758757673418292336292093348, 7.19598013314039086114174891084, 8.100801554629335905381784580980, 9.149156412188436581192825917436, 9.827535245480289108967102730184, 10.21139906970486979508175306560
