| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5·5-s + 6·6-s − 8·8-s + 9·9-s − 10·10-s + 28·11-s − 12·12-s + 86·13-s − 15·15-s + 16·16-s + 66·17-s − 18·18-s + 48·19-s + 20·20-s − 56·22-s + 140·23-s + 24·24-s + 25·25-s − 172·26-s − 27·27-s − 34·29-s + 30·30-s + 284·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.767·11-s − 0.288·12-s + 1.83·13-s − 0.258·15-s + 1/4·16-s + 0.941·17-s − 0.235·18-s + 0.579·19-s + 0.223·20-s − 0.542·22-s + 1.26·23-s + 0.204·24-s + 1/5·25-s − 1.29·26-s − 0.192·27-s − 0.217·29-s + 0.182·30-s + 1.64·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.941945025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.941945025\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 48 T + p^{3} T^{2} \) |
| 23 | \( 1 - 140 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 - 284 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 - 274 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 448 T + p^{3} T^{2} \) |
| 53 | \( 1 + 94 T + p^{3} T^{2} \) |
| 59 | \( 1 + 308 T + p^{3} T^{2} \) |
| 61 | \( 1 + 510 T + p^{3} T^{2} \) |
| 67 | \( 1 + 156 T + p^{3} T^{2} \) |
| 71 | \( 1 - 336 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 - 16 T + p^{3} T^{2} \) |
| 83 | \( 1 + 772 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−9.130844819190144311386925339889, −8.518224113287949639429246775173, −7.52309484429595928373671377914, −6.62400041571631867450575276837, −6.03923168359521793259938463026, −5.22578362778236212234680497517, −3.94074627790389421276066431281, −2.97607428981899776002262588725, −1.41310379533700898382724040282, −0.926330463962524915894946342711,
0.926330463962524915894946342711, 1.41310379533700898382724040282, 2.97607428981899776002262588725, 3.94074627790389421276066431281, 5.22578362778236212234680497517, 6.03923168359521793259938463026, 6.62400041571631867450575276837, 7.52309484429595928373671377914, 8.518224113287949639429246775173, 9.130844819190144311386925339889
