| L(s) = 1 | + 2·3-s − 3·4-s − 2·7-s + 9-s − 6·12-s + 8·13-s + 5·16-s − 4·21-s − 6·25-s − 4·27-s + 6·28-s + 20·31-s − 3·36-s − 12·37-s + 16·39-s + 24·43-s + 10·48-s + 3·49-s − 24·52-s − 2·63-s − 3·64-s + 16·67-s − 16·73-s − 12·75-s + 16·79-s − 11·81-s + 12·84-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 0.755·7-s + 1/3·9-s − 1.73·12-s + 2.21·13-s + 5/4·16-s − 0.872·21-s − 6/5·25-s − 0.769·27-s + 1.13·28-s + 3.59·31-s − 1/2·36-s − 1.97·37-s + 2.56·39-s + 3.65·43-s + 1.44·48-s + 3/7·49-s − 3.32·52-s − 0.251·63-s − 3/8·64-s + 1.95·67-s − 1.87·73-s − 1.38·75-s + 1.80·79-s − 1.22·81-s + 1.30·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.371465887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.371465887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−9.757325112612361159806534316063, −9.502718184597076870574837830250, −8.845360569124199894485226294122, −8.678915900547504955372461266927, −8.184144826454463238531498469573, −7.82867506007281568614416512666, −6.94901622138190974050877028098, −6.09655823991520760248083514453, −5.96443978110022033983848919969, −5.04842931599159887263632672015, −4.07930883034575373640745113861, −3.97700068246488537440823967124, −3.26888344602981049454952325554, −2.48256179865490690419904115851, −1.01248464275202192351966482181,
1.01248464275202192351966482181, 2.48256179865490690419904115851, 3.26888344602981049454952325554, 3.97700068246488537440823967124, 4.07930883034575373640745113861, 5.04842931599159887263632672015, 5.96443978110022033983848919969, 6.09655823991520760248083514453, 6.94901622138190974050877028098, 7.82867506007281568614416512666, 8.184144826454463238531498469573, 8.678915900547504955372461266927, 8.845360569124199894485226294122, 9.502718184597076870574837830250, 9.757325112612361159806534316063
