| L(s) = 1 | − 4·9-s − 12·13-s + 12·17-s − 12·37-s + 4·49-s − 12·53-s + 12·61-s − 4·73-s + 7·81-s + 12·89-s + 20·97-s − 24·101-s + 36·109-s − 12·113-s + 48·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + 163-s + 167-s + 82·169-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 3.32·13-s + 2.91·17-s − 1.97·37-s + 4/7·49-s − 1.64·53-s + 1.53·61-s − 0.468·73-s + 7/9·81-s + 1.27·89-s + 2.03·97-s − 2.38·101-s + 3.44·109-s − 1.12·113-s + 4.43·117-s − 1.27·121-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 − 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.142828243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.142828243\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.077047703461204929084492325796, −7.82052489889028468729194507048, −7.58364156293496430220981160946, −7.17603498857165294526946957767, −6.91206480391201649781115069644, −6.54674558962164947052943554832, −5.89518856138677810376097053587, −5.54075889912839465094842286294, −5.38249751107476733927562752980, −5.18929635940737525380949118409, −4.55532492199595219785990562857, −4.49631810261655333235980996876, −3.45549047916626599500057015030, −3.43791799691771897982384292275, −3.00729198156919321967351433676, −2.60968826564223505142106173077, −2.10565574081574161944434181378, −1.74797791077482380563597014492, −0.860757800917116547926630653998, −0.32293282713193024201633884396,
0.32293282713193024201633884396, 0.860757800917116547926630653998, 1.74797791077482380563597014492, 2.10565574081574161944434181378, 2.60968826564223505142106173077, 3.00729198156919321967351433676, 3.43791799691771897982384292275, 3.45549047916626599500057015030, 4.49631810261655333235980996876, 4.55532492199595219785990562857, 5.18929635940737525380949118409, 5.38249751107476733927562752980, 5.54075889912839465094842286294, 5.89518856138677810376097053587, 6.54674558962164947052943554832, 6.91206480391201649781115069644, 7.17603498857165294526946957767, 7.58364156293496430220981160946, 7.82052489889028468729194507048, 8.077047703461204929084492325796
