| L(s) = 1 | + 4-s − 2·5-s − 5·7-s − 9-s − 2·13-s + 16-s − 2·20-s − 4·23-s − 7·25-s − 5·28-s − 2·29-s + 10·35-s − 36-s + 2·45-s + 14·49-s − 2·52-s − 6·53-s + 8·59-s + 5·63-s + 64-s + 4·65-s + 4·67-s + 12·71-s − 2·80-s − 8·81-s − 24·83-s + 10·91-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 1.88·7-s − 1/3·9-s − 0.554·13-s + 1/4·16-s − 0.447·20-s − 0.834·23-s − 7/5·25-s − 0.944·28-s − 0.371·29-s + 1.69·35-s − 1/6·36-s + 0.298·45-s + 2·49-s − 0.277·52-s − 0.824·53-s + 1.04·59-s + 0.629·63-s + 1/8·64-s + 0.496·65-s + 0.488·67-s + 1.42·71-s − 0.223·80-s − 8/9·81-s − 2.63·83-s + 1.04·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23548 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23548 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−10.29549113238609177901960808588, −9.885566847319717954512995670480, −9.610320735229573674162428613672, −8.891520983856759865543945989227, −8.165524035949405987745903348827, −7.72638365652665252679117085675, −7.06795694550111966917162928355, −6.64812387469283597618551809964, −5.93957381300218141915797314580, −5.53376773666531770794204469723, −4.33408715449728664374263643132, −3.71654578176924070556551344061, −3.13708522456824816788458902897, −2.24060991266744443919987085129, 0,
2.24060991266744443919987085129, 3.13708522456824816788458902897, 3.71654578176924070556551344061, 4.33408715449728664374263643132, 5.53376773666531770794204469723, 5.93957381300218141915797314580, 6.64812387469283597618551809964, 7.06795694550111966917162928355, 7.72638365652665252679117085675, 8.165524035949405987745903348827, 8.891520983856759865543945989227, 9.610320735229573674162428613672, 9.885566847319717954512995670480, 10.29549113238609177901960808588
