| L(s) = 1 | + 9-s − 10·13-s + 12·17-s − 12·29-s − 4·37-s − 13·49-s + 24·53-s − 26·61-s − 4·73-s + 81-s + 14·97-s − 24·101-s − 14·109-s − 24·113-s − 10·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s + 49·169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 2.77·13-s + 2.91·17-s − 2.22·29-s − 0.657·37-s − 1.85·49-s + 3.29·53-s − 3.32·61-s − 0.468·73-s + 1/9·81-s + 1.42·97-s − 2.38·101-s − 1.34·109-s − 2.25·113-s − 0.924·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 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
−8.343202323055163500050210837596, −7.76669311381525406523910155656, −7.65692053420996459201778762115, −7.12643975321476592671795232959, −6.94607178869965744891714265309, −5.83071748809659914225342750428, −5.67959326371779885195862676134, −5.11852008373557425846609857671, −4.74416118025497742788913645945, −3.99631689430066543006110853421, −3.37887843303378899402222918182, −2.87744522522306257344524897509, −2.11983567026486249391714797419, −1.35841755750539912162882790880, 0,
1.35841755750539912162882790880, 2.11983567026486249391714797419, 2.87744522522306257344524897509, 3.37887843303378899402222918182, 3.99631689430066543006110853421, 4.74416118025497742788913645945, 5.11852008373557425846609857671, 5.67959326371779885195862676134, 5.83071748809659914225342750428, 6.94607178869965744891714265309, 7.12643975321476592671795232959, 7.65692053420996459201778762115, 7.76669311381525406523910155656, 8.343202323055163500050210837596
