| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 36·6-s + 233·7-s − 64·8-s + 81·9-s − 498·11-s − 144·12-s + 809·13-s − 932·14-s + 256·16-s − 1.00e3·17-s − 324·18-s − 1.70e3·19-s − 2.09e3·21-s + 1.99e3·22-s + 1.55e3·23-s + 576·24-s − 3.23e3·26-s − 729·27-s + 3.72e3·28-s + 7.83e3·29-s + 977·31-s − 1.02e3·32-s + 4.48e3·33-s + 4.00e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.79·7-s − 0.353·8-s + 1/3·9-s − 1.24·11-s − 0.288·12-s + 1.32·13-s − 1.27·14-s + 1/4·16-s − 0.840·17-s − 0.235·18-s − 1.08·19-s − 1.03·21-s + 0.877·22-s + 0.612·23-s + 0.204·24-s − 0.938·26-s − 0.192·27-s + 0.898·28-s + 1.72·29-s + 0.182·31-s − 0.176·32-s + 0.716·33-s + 0.594·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.318967565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318967565\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 233 T + p^{5} T^{2} \) |
| 11 | \( 1 + 498 T + p^{5} T^{2} \) |
| 13 | \( 1 - 809 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1002 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1705 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1554 T + p^{5} T^{2} \) |
| 29 | \( 1 - 270 p T + p^{5} T^{2} \) |
| 31 | \( 1 - 977 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4822 T + p^{5} T^{2} \) |
| 41 | \( 1 + 8148 T + p^{5} T^{2} \) |
| 43 | \( 1 - 19469 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8418 T + p^{5} T^{2} \) |
| 53 | \( 1 - 17664 T + p^{5} T^{2} \) |
| 59 | \( 1 - 35910 T + p^{5} T^{2} \) |
| 61 | \( 1 - 3527 T + p^{5} T^{2} \) |
| 67 | \( 1 - 57473 T + p^{5} T^{2} \) |
| 71 | \( 1 + 7548 T + p^{5} T^{2} \) |
| 73 | \( 1 + 646 T + p^{5} T^{2} \) |
| 79 | \( 1 + 22720 T + p^{5} T^{2} \) |
| 83 | \( 1 - 11574 T + p^{5} T^{2} \) |
| 89 | \( 1 + 78960 T + p^{5} T^{2} \) |
| 97 | \( 1 - 54593 T + p^{5} T^{2} \) |
| 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
−11.73492518358931884427713116585, −10.86185845060005456212348781334, −10.51703218560129968100816989853, −8.657978306356722578538419262762, −8.181799213384638820050902785890, −6.88033190631482864868791760975, −5.54422624128712596135330302112, −4.43350869263047260774458354276, −2.23049737805514080595466483554, −0.894325545199508713571494779683,
0.894325545199508713571494779683, 2.23049737805514080595466483554, 4.43350869263047260774458354276, 5.54422624128712596135330302112, 6.88033190631482864868791760975, 8.181799213384638820050902785890, 8.657978306356722578538419262762, 10.51703218560129968100816989853, 10.86185845060005456212348781334, 11.73492518358931884427713116585
