| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 36·6-s + 49·7-s + 64·8-s + 81·9-s − 392·11-s − 144·12-s − 142·13-s + 196·14-s + 256·16-s + 2·17-s + 324·18-s + 1.46e3·19-s − 441·21-s − 1.56e3·22-s − 980·23-s − 576·24-s − 568·26-s − 729·27-s + 784·28-s + 3.49e3·29-s − 264·31-s + 1.02e3·32-s + 3.52e3·33-s + 8·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.976·11-s − 0.288·12-s − 0.233·13-s + 0.267·14-s + 1/4·16-s + 0.00167·17-s + 0.235·18-s + 0.932·19-s − 0.218·21-s − 0.690·22-s − 0.386·23-s − 0.204·24-s − 0.164·26-s − 0.192·27-s + 0.188·28-s + 0.771·29-s − 0.0493·31-s + 0.176·32-s + 0.563·33-s + 0.00118·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 11 | \( 1 + 392 T + p^{5} T^{2} \) |
| 13 | \( 1 + 142 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1468 T + p^{5} T^{2} \) |
| 23 | \( 1 + 980 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3494 T + p^{5} T^{2} \) |
| 31 | \( 1 + 264 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2598 T + p^{5} T^{2} \) |
| 41 | \( 1 + 8994 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7388 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2184 T + p^{5} T^{2} \) |
| 53 | \( 1 + 1318 T + p^{5} T^{2} \) |
| 59 | \( 1 - 22932 T + p^{5} T^{2} \) |
| 61 | \( 1 - 19910 T + p^{5} T^{2} \) |
| 67 | \( 1 + 70188 T + p^{5} T^{2} \) |
| 71 | \( 1 - 23996 T + p^{5} T^{2} \) |
| 73 | \( 1 + 70690 T + p^{5} T^{2} \) |
| 79 | \( 1 - 39064 T + p^{5} T^{2} \) |
| 83 | \( 1 + 20676 T + p^{5} T^{2} \) |
| 89 | \( 1 + 46450 T + p^{5} T^{2} \) |
| 97 | \( 1 + 127850 T + p^{5} T^{2} \) |
| 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
−8.637570051184800249379401591720, −7.72809976240000911592421051931, −7.02253591128644716500786281103, −6.01556057284487901235206720441, −5.22364797385504233616940896800, −4.65283425021771943040405056010, −3.47559672680123333962822052528, −2.44952102569067722245769592247, −1.29696739475520241304289375518, 0,
1.29696739475520241304289375518, 2.44952102569067722245769592247, 3.47559672680123333962822052528, 4.65283425021771943040405056010, 5.22364797385504233616940896800, 6.01556057284487901235206720441, 7.02253591128644716500786281103, 7.72809976240000911592421051931, 8.637570051184800249379401591720
