L(s) = 1 | − 3·7-s − 7·13-s − 5·25-s − 18·31-s − 2·37-s − 18·43-s − 49-s + 13·61-s − 21·67-s − 34·73-s + 21·79-s + 21·91-s + 5·97-s − 33·103-s − 4·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯ |
L(s) = 1 | − 1.13·7-s − 1.94·13-s − 25-s − 3.23·31-s − 0.328·37-s − 2.74·43-s − 1/7·49-s + 1.66·61-s − 2.56·67-s − 3.97·73-s + 2.36·79-s + 2.20·91-s + 0.507·97-s − 3.25·103-s − 0.383·109-s + 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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.367442730773255774982940818209, −9.318200144289380673419647218715, −8.697123802597151813263439394246, −8.276757127874665655284573368061, −7.63190557378279379830316841925, −7.44002287338258914107594420830, −6.86412660394563762024659658038, −6.85897222873525122875098841619, −6.05182093103421918018518510515, −5.70186709727430272841877203075, −5.21800233738994317070411948060, −4.89764861175858715021026909054, −4.19968525243674936408754935089, −3.76384018313927621056478477213, −3.16206579279057675422208939274, −2.88555062163160274369068024720, −1.97807547862772260725466015185, −1.70950401533280722758584861901, 0, 0,
1.70950401533280722758584861901, 1.97807547862772260725466015185, 2.88555062163160274369068024720, 3.16206579279057675422208939274, 3.76384018313927621056478477213, 4.19968525243674936408754935089, 4.89764861175858715021026909054, 5.21800233738994317070411948060, 5.70186709727430272841877203075, 6.05182093103421918018518510515, 6.85897222873525122875098841619, 6.86412660394563762024659658038, 7.44002287338258914107594420830, 7.63190557378279379830316841925, 8.276757127874665655284573368061, 8.697123802597151813263439394246, 9.318200144289380673419647218715, 9.367442730773255774982940818209