| L(s) = 1 | + 2·2-s + 4·4-s − 6·5-s − 16·7-s + 8·8-s − 12·10-s − 12·11-s + 38·13-s − 32·14-s + 16·16-s + 126·17-s + 20·19-s − 24·20-s − 24·22-s − 168·23-s − 89·25-s + 76·26-s − 64·28-s − 30·29-s − 88·31-s + 32·32-s + 252·34-s + 96·35-s + 254·37-s + 40·38-s − 48·40-s − 42·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.536·5-s − 0.863·7-s + 0.353·8-s − 0.379·10-s − 0.328·11-s + 0.810·13-s − 0.610·14-s + 1/4·16-s + 1.79·17-s + 0.241·19-s − 0.268·20-s − 0.232·22-s − 1.52·23-s − 0.711·25-s + 0.573·26-s − 0.431·28-s − 0.192·29-s − 0.509·31-s + 0.176·32-s + 1.27·34-s + 0.463·35-s + 1.12·37-s + 0.170·38-s − 0.189·40-s − 0.159·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.339937723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.339937723\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 538 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 + 792 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
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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
−18.44187078213435321181228660570, −16.49283416198810241904062500247, −15.69789007701018280405867310968, −14.21619922753104885303434400485, −12.88299997955728899305459192822, −11.69135476711267912580710906154, −9.997177907882694374396051296820, −7.79784374777438716958608823195, −5.91841366397310679443986977087, −3.60929026962551384976945217111,
3.60929026962551384976945217111, 5.91841366397310679443986977087, 7.79784374777438716958608823195, 9.997177907882694374396051296820, 11.69135476711267912580710906154, 12.88299997955728899305459192822, 14.21619922753104885303434400485, 15.69789007701018280405867310968, 16.49283416198810241904062500247, 18.44187078213435321181228660570
