This L-function arises from both an elliptic curve over a number field,
and from a genus 2 curve over the rationals. It is probably the L-function
of smallest conductor with that property.
| L(s) = 1 | − 2·4-s − 2·7-s − 2·9-s + 4·16-s + 5·17-s − 6·23-s + 2·25-s + 4·28-s − 2·31-s + 4·36-s + 12·47-s − 2·49-s + 4·63-s − 8·64-s − 10·68-s + 6·71-s − 20·73-s − 2·79-s − 5·81-s + 12·92-s − 8·97-s − 4·100-s − 8·103-s − 8·112-s + 24·113-s − 10·119-s − 10·121-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s − 2/3·9-s + 16-s + 1.21·17-s − 1.25·23-s + 2/5·25-s + 0.755·28-s − 0.359·31-s + 2/3·36-s + 1.75·47-s − 2/7·49-s + 0.503·63-s − 64-s − 1.21·68-s + 0.712·71-s − 2.34·73-s − 0.225·79-s − 5/9·81-s + 1.25·92-s − 0.812·97-s − 2/5·100-s − 0.788·103-s − 0.755·112-s + 2.25·113-s − 0.916·119-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4366701692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4366701692\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−14.19044829739582126239819958846, −13.54189737490713903220419819081, −12.86589430836834663288318270965, −12.29726535287971277212019804411, −11.76221275592136857524468249377, −10.71210139881204999206150662240, −10.05681574304479561828369141584, −9.514581676057178887948389923580, −8.766450619065699860669585506452, −8.104427830147069626222932574670, −7.25728763856226074637060722968, −6.01912712781998810706342352491, −5.48237499609119068897650555264, −4.21524744281209647906717432747, −3.17400203367696839713552335058,
3.17400203367696839713552335058, 4.21524744281209647906717432747, 5.48237499609119068897650555264, 6.01912712781998810706342352491, 7.25728763856226074637060722968, 8.104427830147069626222932574670, 8.766450619065699860669585506452, 9.514581676057178887948389923580, 10.05681574304479561828369141584, 10.71210139881204999206150662240, 11.76221275592136857524468249377, 12.29726535287971277212019804411, 12.86589430836834663288318270965, 13.54189737490713903220419819081, 14.19044829739582126239819958846
