| L(s) = 1 | + 2·5-s + 3·9-s + 12·13-s − 2·17-s + 5·25-s − 20·29-s + 2·37-s + 20·41-s + 6·45-s − 14·53-s + 10·61-s + 24·65-s + 6·73-s − 4·85-s − 10·89-s + 36·97-s + 2·101-s − 6·109-s − 28·113-s + 36·117-s + 11·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 9-s + 3.32·13-s − 0.485·17-s + 25-s − 3.71·29-s + 0.328·37-s + 3.12·41-s + 0.894·45-s − 1.92·53-s + 1.28·61-s + 2.97·65-s + 0.702·73-s − 0.433·85-s − 1.05·89-s + 3.65·97-s + 0.199·101-s − 0.574·109-s − 2.63·113-s + 3.32·117-s + 121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.928677610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.928677610\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 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
−9.596217089251235973144226146753, −9.270842932231006894609393718444, −8.816278662756817364832643584961, −8.690156043388332829496223141848, −7.912591218713821630998378719785, −7.74772030784588976463769790506, −7.20508973447611266448702088031, −6.75283449306363278982894449113, −6.22360093555634451850504371113, −6.00123654017432615011133482722, −5.73116554112964815885542667805, −5.24281242106899192112193828474, −4.47188546696729274756658672390, −4.12494590271738282204569281943, −3.61633248902851439760982642349, −3.42891139092492171791587163282, −2.47314716694803096831033800656, −1.86345545908322150849026141964, −1.45286753045269302624972575186, −0.854463060309764483263012365521,
0.854463060309764483263012365521, 1.45286753045269302624972575186, 1.86345545908322150849026141964, 2.47314716694803096831033800656, 3.42891139092492171791587163282, 3.61633248902851439760982642349, 4.12494590271738282204569281943, 4.47188546696729274756658672390, 5.24281242106899192112193828474, 5.73116554112964815885542667805, 6.00123654017432615011133482722, 6.22360093555634451850504371113, 6.75283449306363278982894449113, 7.20508973447611266448702088031, 7.74772030784588976463769790506, 7.912591218713821630998378719785, 8.690156043388332829496223141848, 8.816278662756817364832643584961, 9.270842932231006894609393718444, 9.596217089251235973144226146753
