| L(s) = 1 | + 5-s + 7-s + 2·13-s − 2·17-s + 4·19-s + 25-s − 2·29-s + 4·31-s + 35-s + 2·37-s + 2·41-s + 8·43-s + 4·47-s + 49-s + 2·53-s − 12·59-s − 14·61-s + 2·65-s + 8·67-s + 8·71-s − 2·73-s + 8·79-s + 4·83-s − 2·85-s + 18·89-s + 2·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s + 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s − 1.79·61-s + 0.248·65-s + 0.977·67-s + 0.949·71-s − 0.234·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s + 1.90·89-s + 0.209·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.178609352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178609352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−9.095367825196243394897043664297, −8.079440801261388063720120357582, −7.48918360251945918270184807870, −6.50277156323105969838700776847, −5.84778792500317935629611203265, −5.00140669202665528815724979200, −4.16143909826850275013312298886, −3.11863300382156819629828204699, −2.10532609343856631936221825630, −0.982468938855557328742590710518,
0.982468938855557328742590710518, 2.10532609343856631936221825630, 3.11863300382156819629828204699, 4.16143909826850275013312298886, 5.00140669202665528815724979200, 5.84778792500317935629611203265, 6.50277156323105969838700776847, 7.48918360251945918270184807870, 8.079440801261388063720120357582, 9.095367825196243394897043664297
