| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 5·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 10·10-s + 16·11-s − 12·12-s + 58·13-s − 14·14-s − 15·15-s + 16·16-s + 34·17-s + 18·18-s + 64·19-s + 20·20-s + 21·21-s + 32·22-s − 16·23-s − 24·24-s + 25·25-s + 116·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.438·11-s − 0.288·12-s + 1.23·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.772·19-s + 0.223·20-s + 0.218·21-s + 0.310·22-s − 0.145·23-s − 0.204·24-s + 1/5·25-s + 0.874·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.561377694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.561377694\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 64 T + p^{3} T^{2} \) |
| 23 | \( 1 + 16 T + p^{3} T^{2} \) |
| 29 | \( 1 - 62 T + p^{3} T^{2} \) |
| 31 | \( 1 - 60 T + p^{3} T^{2} \) |
| 37 | \( 1 - 150 T + p^{3} T^{2} \) |
| 41 | \( 1 - 474 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 662 T + p^{3} T^{2} \) |
| 59 | \( 1 + 324 T + p^{3} T^{2} \) |
| 61 | \( 1 + 514 T + p^{3} T^{2} \) |
| 67 | \( 1 + 372 T + p^{3} T^{2} \) |
| 71 | \( 1 + 412 T + p^{3} T^{2} \) |
| 73 | \( 1 + 770 T + p^{3} T^{2} \) |
| 79 | \( 1 + 560 T + p^{3} T^{2} \) |
| 83 | \( 1 + 852 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1466 T + p^{3} T^{2} \) |
| 97 | \( 1 + 178 T + p^{3} T^{2} \) |
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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
−11.97364056434371902318750708257, −11.12449009658033979132821663504, −10.16441221620499439145078645730, −9.096018975774828468513808346979, −7.63277794589785215231364822472, −6.35795595471286948298806532584, −5.79219516591701206102374658338, −4.44567753854291893774718330927, −3.15019099597829607153463395613, −1.27465376451361724580241562803,
1.27465376451361724580241562803, 3.15019099597829607153463395613, 4.44567753854291893774718330927, 5.79219516591701206102374658338, 6.35795595471286948298806532584, 7.63277794589785215231364822472, 9.096018975774828468513808346979, 10.16441221620499439145078645730, 11.12449009658033979132821663504, 11.97364056434371902318750708257
