| L(s) = 1 | + 4-s + 16-s − 7·25-s + 4·37-s + 16·43-s + 64-s + 4·67-s − 2·79-s − 7·100-s + 4·109-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 16·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/4·16-s − 7/5·25-s + 0.657·37-s + 2.43·43-s + 1/8·64-s + 0.488·67-s − 0.225·79-s − 0.699·100-s + 0.383·109-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-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 + 1.21·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.129683297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.129683297\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−8.040849006795850625744290490432, −7.86433970519498838051603991596, −7.41980752467687128419459320216, −6.95945540478754831177732177379, −6.46656879798063291179868351061, −5.94371449806009053495319795057, −5.69764611578402353089760755346, −5.15428087704295681166595776854, −4.39086112686115727830378055155, −4.11097932559462224118976384901, −3.48918985769468073275271611503, −2.82305847242492586564742785018, −2.30289991562456535356218231988, −1.67505854293018902975009387356, −0.70830640819645163144339448768,
0.70830640819645163144339448768, 1.67505854293018902975009387356, 2.30289991562456535356218231988, 2.82305847242492586564742785018, 3.48918985769468073275271611503, 4.11097932559462224118976384901, 4.39086112686115727830378055155, 5.15428087704295681166595776854, 5.69764611578402353089760755346, 5.94371449806009053495319795057, 6.46656879798063291179868351061, 6.95945540478754831177732177379, 7.41980752467687128419459320216, 7.86433970519498838051603991596, 8.040849006795850625744290490432
