| L(s) = 1 | + 2·2-s − 3·3-s − 4·4-s − 6·6-s + 7·7-s − 24·8-s + 9·9-s − 21·11-s + 12·12-s − 24·13-s + 14·14-s − 16·16-s + 22·17-s + 18·18-s + 16·19-s − 21·21-s − 42·22-s + 25·23-s + 72·24-s − 48·26-s − 27·27-s − 28·28-s + 167·29-s + 10·31-s + 160·32-s + 63·33-s + 44·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.575·11-s + 0.288·12-s − 0.512·13-s + 0.267·14-s − 1/4·16-s + 0.313·17-s + 0.235·18-s + 0.193·19-s − 0.218·21-s − 0.407·22-s + 0.226·23-s + 0.612·24-s − 0.362·26-s − 0.192·27-s − 0.188·28-s + 1.06·29-s + 0.0579·31-s + 0.883·32-s + 0.332·33-s + 0.221·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.642064131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642064131\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 + 21 T + p^{3} T^{2} \) |
| 13 | \( 1 + 24 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 - 25 T + p^{3} T^{2} \) |
| 29 | \( 1 - 167 T + p^{3} T^{2} \) |
| 31 | \( 1 - 10 T + p^{3} T^{2} \) |
| 37 | \( 1 - 133 T + p^{3} T^{2} \) |
| 41 | \( 1 + 168 T + p^{3} T^{2} \) |
| 43 | \( 1 - 97 T + p^{3} T^{2} \) |
| 47 | \( 1 - 400 T + p^{3} T^{2} \) |
| 53 | \( 1 - 182 T + p^{3} T^{2} \) |
| 59 | \( 1 - 488 T + p^{3} T^{2} \) |
| 61 | \( 1 - 28 T + p^{3} T^{2} \) |
| 67 | \( 1 - 967 T + p^{3} T^{2} \) |
| 71 | \( 1 + 285 T + p^{3} T^{2} \) |
| 73 | \( 1 - 838 T + p^{3} T^{2} \) |
| 79 | \( 1 + 469 T + p^{3} T^{2} \) |
| 83 | \( 1 - 406 T + p^{3} T^{2} \) |
| 89 | \( 1 - 324 T + p^{3} T^{2} \) |
| 97 | \( 1 - 114 T + p^{3} T^{2} \) |
| 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
−10.47962296542795952564576634696, −9.703699336488039132238434977987, −8.658707706382875618606886022389, −7.70311081632331248256716098195, −6.56232493399649424470273919119, −5.48155291614182651935287692198, −4.92085966244524190500357276138, −3.91398616866517065064944274955, −2.59610125188529871055992797638, −0.72887306247798585303484237221,
0.72887306247798585303484237221, 2.59610125188529871055992797638, 3.91398616866517065064944274955, 4.92085966244524190500357276138, 5.48155291614182651935287692198, 6.56232493399649424470273919119, 7.70311081632331248256716098195, 8.658707706382875618606886022389, 9.703699336488039132238434977987, 10.47962296542795952564576634696
