| L(s) = 1 | + 27·3-s + 125·5-s − 512·7-s + 729·9-s − 5.46e3·11-s + 1.01e4·13-s + 3.37e3·15-s − 9.91e3·17-s + 1.24e4·19-s − 1.38e4·21-s − 3.36e4·23-s + 1.56e4·25-s + 1.96e4·27-s − 1.87e5·29-s + 4.25e4·31-s − 1.47e5·33-s − 6.40e4·35-s − 5.44e5·37-s + 2.74e5·39-s + 3.74e5·41-s + 5.40e5·43-s + 9.11e4·45-s − 1.33e6·47-s − 5.61e5·49-s − 2.67e5·51-s + 1.30e6·53-s − 6.82e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.564·7-s + 1/3·9-s − 1.23·11-s + 1.28·13-s + 0.258·15-s − 0.489·17-s + 0.415·19-s − 0.325·21-s − 0.575·23-s + 1/5·25-s + 0.192·27-s − 1.43·29-s + 0.256·31-s − 0.714·33-s − 0.252·35-s − 1.76·37-s + 0.740·39-s + 0.848·41-s + 1.03·43-s + 0.149·45-s − 1.88·47-s − 0.681·49-s − 0.282·51-s + 1.20·53-s − 0.553·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| good | 7 | \( 1 + 512 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5460 T + p^{7} T^{2} \) |
| 13 | \( 1 - 782 p T + p^{7} T^{2} \) |
| 17 | \( 1 + 9918 T + p^{7} T^{2} \) |
| 19 | \( 1 - 12436 T + p^{7} T^{2} \) |
| 23 | \( 1 + 33600 T + p^{7} T^{2} \) |
| 29 | \( 1 + 187914 T + p^{7} T^{2} \) |
| 31 | \( 1 - 42592 T + p^{7} T^{2} \) |
| 37 | \( 1 + 544066 T + p^{7} T^{2} \) |
| 41 | \( 1 - 374394 T + p^{7} T^{2} \) |
| 43 | \( 1 - 540532 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1338360 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1308222 T + p^{7} T^{2} \) |
| 59 | \( 1 + 262740 T + p^{7} T^{2} \) |
| 61 | \( 1 + 976330 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3559172 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2673720 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3032134 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5475808 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2231556 T + p^{7} T^{2} \) |
| 89 | \( 1 + 10050678 T + p^{7} T^{2} \) |
| 97 | \( 1 - 5727554 T + p^{7} T^{2} \) |
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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
−10.32737482471637215150734159373, −9.394815471197496491384459784730, −8.498405629393866490786675704120, −7.50797341354544861232999962217, −6.32041948911103452292749135801, −5.32315204527691294066163117124, −3.84622825285153355522476492189, −2.80585642508887432164779339835, −1.61603332443757290603817402412, 0,
1.61603332443757290603817402412, 2.80585642508887432164779339835, 3.84622825285153355522476492189, 5.32315204527691294066163117124, 6.32041948911103452292749135801, 7.50797341354544861232999962217, 8.498405629393866490786675704120, 9.394815471197496491384459784730, 10.32737482471637215150734159373
