| L(s) = 1 | + 2-s − 4-s − 3·7-s − 3·8-s − 11-s + 2·13-s − 3·14-s − 16-s + 3·17-s − 19-s − 22-s + 23-s + 2·26-s + 3·28-s − 6·29-s + 4·31-s + 5·32-s + 3·34-s + 37-s − 38-s + 5·41-s + 4·43-s + 44-s + 46-s + 3·47-s + 2·49-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.13·7-s − 1.06·8-s − 0.301·11-s + 0.554·13-s − 0.801·14-s − 1/4·16-s + 0.727·17-s − 0.229·19-s − 0.213·22-s + 0.208·23-s + 0.392·26-s + 0.566·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s + 0.514·34-s + 0.164·37-s − 0.162·38-s + 0.780·41-s + 0.609·43-s + 0.150·44-s + 0.147·46-s + 0.437·47-s + 2/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584153231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584153231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−9.045139892119569417137431279678, −8.219780045255613256650843886676, −7.31312039922101041649638259142, −6.32130068262525770566739304029, −5.81623861250261361288562056953, −5.00870377288382819137628081166, −3.96790742024516625672803719382, −3.42923538887095862608540181613, −2.49582699915333217995686926465, −0.70855844396652026469149918924,
0.70855844396652026469149918924, 2.49582699915333217995686926465, 3.42923538887095862608540181613, 3.96790742024516625672803719382, 5.00870377288382819137628081166, 5.81623861250261361288562056953, 6.32130068262525770566739304029, 7.31312039922101041649638259142, 8.219780045255613256650843886676, 9.045139892119569417137431279678
