| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s + 227·7-s − 64·8-s + 81·9-s − 100·10-s − 303·11-s − 144·12-s + 169·13-s − 908·14-s − 225·15-s + 256·16-s − 1.20e3·17-s − 324·18-s − 34·19-s + 400·20-s − 2.04e3·21-s + 1.21e3·22-s − 4.39e3·23-s + 576·24-s + 625·25-s − 676·26-s − 729·27-s + 3.63e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.75·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.755·11-s − 0.288·12-s + 0.277·13-s − 1.23·14-s − 0.258·15-s + 1/4·16-s − 1.00·17-s − 0.235·18-s − 0.0216·19-s + 0.223·20-s − 1.01·21-s + 0.533·22-s − 1.73·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.875·28-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 | \( 1 - 227 T + p^{5} T^{2} \) |
| 11 | \( 1 + 303 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1203 T + p^{5} T^{2} \) |
| 19 | \( 1 + 34 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4395 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3246 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1778 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7819 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10659 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7876 T + p^{5} T^{2} \) |
| 47 | \( 1 - 21102 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9723 T + p^{5} T^{2} \) |
| 59 | \( 1 + 50520 T + p^{5} T^{2} \) |
| 61 | \( 1 + 23599 T + p^{5} T^{2} \) |
| 67 | \( 1 - 55484 T + p^{5} T^{2} \) |
| 71 | \( 1 - 20721 T + p^{5} T^{2} \) |
| 73 | \( 1 + 40042 T + p^{5} T^{2} \) |
| 79 | \( 1 + 82753 T + p^{5} T^{2} \) |
| 83 | \( 1 + 42048 T + p^{5} T^{2} \) |
| 89 | \( 1 - 39441 T + p^{5} T^{2} \) |
| 97 | \( 1 - 151667 T + p^{5} T^{2} \) |
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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
−10.25943416542025317562638930922, −9.057280527188201484480309364247, −8.154111802280184335549480788975, −7.45418062661972412359448304870, −6.17751310936590689285132639152, −5.28517510480140860563302734527, −4.26659571281445905722748528791, −2.27506659431639832601701869565, −1.46336511794818534234419729551, 0,
1.46336511794818534234419729551, 2.27506659431639832601701869565, 4.26659571281445905722748528791, 5.28517510480140860563302734527, 6.17751310936590689285132639152, 7.45418062661972412359448304870, 8.154111802280184335549480788975, 9.057280527188201484480309364247, 10.25943416542025317562638930922
