L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s − 4·16-s − 12·23-s + 4·27-s − 6·31-s + 4·33-s − 4·37-s − 4·47-s − 8·48-s − 5·49-s + 8·53-s − 20·59-s + 6·67-s − 24·69-s − 16·71-s + 5·81-s − 12·93-s − 34·97-s + 6·99-s + 8·103-s − 8·111-s + 8·113-s − 7·121-s + 127-s + 131-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s + 0.603·11-s − 16-s − 2.50·23-s + 0.769·27-s − 1.07·31-s + 0.696·33-s − 0.657·37-s − 0.583·47-s − 1.15·48-s − 5/7·49-s + 1.09·53-s − 2.60·59-s + 0.733·67-s − 2.88·69-s − 1.89·71-s + 5/9·81-s − 1.24·93-s − 3.45·97-s + 0.603·99-s + 0.788·103-s − 0.759·111-s + 0.752·113-s − 0.636·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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
−8.114757177440921419069207299350, −7.74300600574245161145141332474, −7.34923234321660315078497154764, −6.77287251257420576381216593643, −6.47550891297445546652062110512, −5.83602188561825449212965209319, −5.44264591099408884558544576394, −4.48018036005395779311676580836, −4.38637866918120839883207984982, −3.75734065175266443138175607180, −3.28631221908682607371983175186, −2.62960787106095443058504699500, −1.90319955654708104595670502735, −1.60835528081174442082226795369, 0,
1.60835528081174442082226795369, 1.90319955654708104595670502735, 2.62960787106095443058504699500, 3.28631221908682607371983175186, 3.75734065175266443138175607180, 4.38637866918120839883207984982, 4.48018036005395779311676580836, 5.44264591099408884558544576394, 5.83602188561825449212965209319, 6.47550891297445546652062110512, 6.77287251257420576381216593643, 7.34923234321660315078497154764, 7.74300600574245161145141332474, 8.114757177440921419069207299350