| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 2·13-s − 15-s − 2·17-s + 2·21-s + 2·23-s + 25-s + 27-s − 4·29-s + 4·31-s − 2·35-s + 2·37-s + 2·39-s + 4·41-s − 10·43-s − 45-s − 6·47-s − 3·49-s − 2·51-s + 6·53-s − 4·59-s − 10·61-s + 2·63-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.320·39-s + 0.624·41-s − 1.52·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.520·59-s − 1.28·61-s + 0.251·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.10494860585205, −13.71738350319143, −13.17917640152698, −12.81718743742997, −12.15640344827018, −11.50627968958853, −11.35064756676668, −10.70405453666816, −10.22352189383803, −9.587688263673400, −9.028683874539496, −8.580214703913577, −8.161168683997981, −7.614107207429696, −7.233096609712540, −6.439657064506731, −6.099820117997387, −5.158589016211476, −4.767552473532970, −4.220120196203279, −3.559291685789429, −3.081023805656040, −2.312349794880842, −1.662535830899125, −1.040799928106296, 0,
1.040799928106296, 1.662535830899125, 2.312349794880842, 3.081023805656040, 3.559291685789429, 4.220120196203279, 4.767552473532970, 5.158589016211476, 6.099820117997387, 6.439657064506731, 7.233096609712540, 7.614107207429696, 8.161168683997981, 8.580214703913577, 9.028683874539496, 9.587688263673400, 10.22352189383803, 10.70405453666816, 11.35064756676668, 11.50627968958853, 12.15640344827018, 12.81718743742997, 13.17917640152698, 13.71738350319143, 14.10494860585205
