| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 9-s + 6·11-s − 6·12-s + 5·16-s − 2·18-s − 12·22-s + 8·24-s + 8·25-s + 4·27-s + 12·29-s − 8·31-s − 6·32-s − 12·33-s + 3·36-s + 4·37-s − 12·41-s + 18·44-s − 10·48-s − 4·49-s − 16·50-s − 8·54-s − 24·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 1/3·9-s + 1.80·11-s − 1.73·12-s + 5/4·16-s − 0.471·18-s − 2.55·22-s + 1.63·24-s + 8/5·25-s + 0.769·27-s + 2.22·29-s − 1.43·31-s − 1.06·32-s − 2.08·33-s + 1/2·36-s + 0.657·37-s − 1.87·41-s + 2.71·44-s − 1.44·48-s − 4/7·49-s − 2.26·50-s − 1.08·54-s − 3.15·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3367331366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3367331366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.30556197279685379006222456182, −14.54992152661329495575061204561, −14.41827408583761118588371976444, −13.44543812407287555328246665904, −12.43098421148282694170278084302, −12.25214400689579708503436217425, −11.54499026212327286047161335028, −11.27519791264757477370321920144, −10.56781245943628211088939549144, −10.12875466226042731418245521379, −9.409888699472237083423453301165, −8.723552398928833326275274416666, −8.460277818044254713677999124030, −7.32114391188213531162876555526, −6.56861176456089163521667146027, −6.50789825526095172594949771550, −5.45095385276715594437220529329, −4.47633725444255445471405884856, −3.10679946474497838366817225872, −1.29388794649529126458273631718,
1.29388794649529126458273631718, 3.10679946474497838366817225872, 4.47633725444255445471405884856, 5.45095385276715594437220529329, 6.50789825526095172594949771550, 6.56861176456089163521667146027, 7.32114391188213531162876555526, 8.460277818044254713677999124030, 8.723552398928833326275274416666, 9.409888699472237083423453301165, 10.12875466226042731418245521379, 10.56781245943628211088939549144, 11.27519791264757477370321920144, 11.54499026212327286047161335028, 12.25214400689579708503436217425, 12.43098421148282694170278084302, 13.44543812407287555328246665904, 14.41827408583761118588371976444, 14.54992152661329495575061204561, 15.30556197279685379006222456182
