| L(s) = 1 | − 5·4-s − 8·5-s − 10·7-s + 20·11-s + 9·16-s − 30·17-s − 12·19-s + 40·20-s − 70·23-s − 2·25-s + 50·28-s + 80·35-s − 40·43-s − 100·44-s − 20·47-s − 23·49-s − 160·55-s − 80·61-s + 35·64-s + 150·68-s + 210·73-s + 60·76-s − 200·77-s − 72·80-s + 80·83-s + 240·85-s + 350·92-s + ⋯ |
| L(s) = 1 | − 5/4·4-s − 8/5·5-s − 1.42·7-s + 1.81·11-s + 9/16·16-s − 1.76·17-s − 0.631·19-s + 2·20-s − 3.04·23-s − 0.0799·25-s + 1.78·28-s + 16/7·35-s − 0.930·43-s − 2.27·44-s − 0.425·47-s − 0.469·49-s − 2.90·55-s − 1.31·61-s + 0.546·64-s + 2.20·68-s + 2.87·73-s + 0.789·76-s − 2.59·77-s − 0.899·80-s + 0.963·83-s + 2.82·85-s + 3.80·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02104612688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02104612688\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 12 T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 p T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 15 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1357 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 622 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2270 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 115 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6637 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7405 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 92 T + p^{2} T^{2} )( 1 + 92 T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 105 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11182 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 3790 T^{2} + p^{4} T^{4} \) |
| 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
−13.60524041461021130457958073938, −12.66371439073440530966357064368, −11.84960236131811649485396635033, −11.46411601605849149334951535598, −10.91677493390806239101244463563, −9.876854004928482671919582741511, −9.875307009138764980751996367538, −9.118088575752681824176868894902, −8.862464216143561582021478303180, −8.055517691702938353805255156018, −7.938642796243191723131636863330, −6.88578829635368436837022156261, −6.31264430671184852908176828513, −6.20960503546652108281345506545, −4.83281403600490277107489784171, −4.23796266244374398250605457553, −3.73080687906166294458054687854, −3.68771314640784669052715457123, −2.04984020821034493401802594745, −0.087293086882517839292025294900,
0.087293086882517839292025294900, 2.04984020821034493401802594745, 3.68771314640784669052715457123, 3.73080687906166294458054687854, 4.23796266244374398250605457553, 4.83281403600490277107489784171, 6.20960503546652108281345506545, 6.31264430671184852908176828513, 6.88578829635368436837022156261, 7.938642796243191723131636863330, 8.055517691702938353805255156018, 8.862464216143561582021478303180, 9.118088575752681824176868894902, 9.875307009138764980751996367538, 9.876854004928482671919582741511, 10.91677493390806239101244463563, 11.46411601605849149334951535598, 11.84960236131811649485396635033, 12.66371439073440530966357064368, 13.60524041461021130457958073938
