| 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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 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 | \( 1 + 54593 T + p^{5} T^{2} \) |
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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
−12.05377979490999764697099329200, −10.17631816897608523031984607405, −9.960509751041248555248845593265, −8.336502984152169202259132925214, −7.16188647872082414105910310476, −6.15713671492277856472731179111, −4.77064804491440810237052412241, −3.28450931778815424476715045255, −2.48960984296575296368818210415, 0,
2.48960984296575296368818210415, 3.28450931778815424476715045255, 4.77064804491440810237052412241, 6.15713671492277856472731179111, 7.16188647872082414105910310476, 8.336502984152169202259132925214, 9.960509751041248555248845593265, 10.17631816897608523031984607405, 12.05377979490999764697099329200
