L(s) = 1 | − 2·4-s + 4·13-s + 4·16-s + 25-s + 4·37-s + 8·49-s − 8·52-s + 16·61-s − 8·64-s + 28·73-s − 20·97-s − 2·100-s − 8·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 4-s + 1.10·13-s + 16-s + 1/5·25-s + 0.657·37-s + 8/7·49-s − 1.10·52-s + 2.04·61-s − 64-s + 3.27·73-s − 2.03·97-s − 1/5·100-s − 0.766·109-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.038842612\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038842612\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.42622977065269933855893812010, −9.813157066547424337510185330525, −9.429118435037894227504984756223, −8.878904432761376537547579456374, −8.288828315301589688901603419147, −8.091937983271484055936897365069, −7.23363581552713227021582441250, −6.61668149635008544378180724561, −5.94171836444501009690226869744, −5.39504256604270189333590479848, −4.77867221017145857963863619851, −3.93826359666462119089513224541, −3.60167852700477778234866407088, −2.46590351451006153706317864020, −1.05279196353716811796825860551,
1.05279196353716811796825860551, 2.46590351451006153706317864020, 3.60167852700477778234866407088, 3.93826359666462119089513224541, 4.77867221017145857963863619851, 5.39504256604270189333590479848, 5.94171836444501009690226869744, 6.61668149635008544378180724561, 7.23363581552713227021582441250, 8.091937983271484055936897365069, 8.288828315301589688901603419147, 8.878904432761376537547579456374, 9.429118435037894227504984756223, 9.813157066547424337510185330525, 10.42622977065269933855893812010