| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 25·5-s + 36·6-s − 106·7-s + 64·8-s + 81·9-s − 100·10-s + 226·11-s + 144·12-s − 169·13-s − 424·14-s − 225·15-s + 256·16-s − 898·17-s + 324·18-s − 1.57e3·19-s − 400·20-s − 954·21-s + 904·22-s + 1.50e3·23-s + 576·24-s + 625·25-s − 676·26-s + 729·27-s − 1.69e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.817·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.563·11-s + 0.288·12-s − 0.277·13-s − 0.578·14-s − 0.258·15-s + 1/4·16-s − 0.753·17-s + 0.235·18-s − 1.00·19-s − 0.223·20-s − 0.472·21-s + 0.398·22-s + 0.591·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.408·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 106 T + p^{5} T^{2} \) |
| 11 | \( 1 - 226 T + p^{5} T^{2} \) |
| 17 | \( 1 + 898 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1578 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1500 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7534 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5286 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4142 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14150 T + p^{5} T^{2} \) |
| 43 | \( 1 + 11056 T + p^{5} T^{2} \) |
| 47 | \( 1 + 15042 T + p^{5} T^{2} \) |
| 53 | \( 1 + 15730 T + p^{5} T^{2} \) |
| 59 | \( 1 + 10046 T + p^{5} T^{2} \) |
| 61 | \( 1 - 19358 T + p^{5} T^{2} \) |
| 67 | \( 1 + 966 T + p^{5} T^{2} \) |
| 71 | \( 1 - 24382 T + p^{5} T^{2} \) |
| 73 | \( 1 + 11078 T + p^{5} T^{2} \) |
| 79 | \( 1 + 31372 T + p^{5} T^{2} \) |
| 83 | \( 1 + 8838 T + p^{5} T^{2} \) |
| 89 | \( 1 - 46354 T + p^{5} T^{2} \) |
| 97 | \( 1 - 42770 T + p^{5} T^{2} \) |
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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
−10.03916646723252955792615394154, −9.098733407768856836581933613891, −8.163839409526349182557035370690, −6.97150106043129918701599787315, −6.39215326890455535503818427414, −4.92977716346190147677371498491, −3.90721858820884818044166153425, −3.07832125072038112242270257760, −1.82094718903775750735701033165, 0,
1.82094718903775750735701033165, 3.07832125072038112242270257760, 3.90721858820884818044166153425, 4.92977716346190147677371498491, 6.39215326890455535503818427414, 6.97150106043129918701599787315, 8.163839409526349182557035370690, 9.098733407768856836581933613891, 10.03916646723252955792615394154
