L(s) = 1 | + 4-s − 5·7-s + 6·11-s + 16-s + 2·25-s − 5·28-s − 7·37-s + 3·41-s + 6·44-s + 6·47-s + 7·49-s + 64-s + 7·67-s − 9·71-s + 13·73-s − 30·77-s + 24·83-s + 2·100-s + 15·101-s − 5·112-s + 14·121-s + 127-s + 131-s + 137-s + 139-s − 7·148-s + 149-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 1.88·7-s + 1.80·11-s + 1/4·16-s + 2/5·25-s − 0.944·28-s − 1.15·37-s + 0.468·41-s + 0.904·44-s + 0.875·47-s + 49-s + 1/8·64-s + 0.855·67-s − 1.06·71-s + 1.52·73-s − 3.41·77-s + 2.63·83-s + 1/5·100-s + 1.49·101-s − 0.472·112-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.575·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.752329225\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.752329225\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
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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
−8.820569452891575312035096497533, −8.139743426164762426896802064442, −7.51577421793174998945463067678, −7.08435344514735874699303383882, −6.61926369665901221466870868365, −6.36198654373467615234893955818, −6.10486610480185424206658058416, −5.36588369481349339860294761879, −4.78479566635529712614734648898, −3.90006589527511075765360779631, −3.70685499028521781137005262636, −3.17349333332914601855466930718, −2.51293610138926031796492183564, −1.69688687831240788452505004580, −0.72582147763788633757709097494,
0.72582147763788633757709097494, 1.69688687831240788452505004580, 2.51293610138926031796492183564, 3.17349333332914601855466930718, 3.70685499028521781137005262636, 3.90006589527511075765360779631, 4.78479566635529712614734648898, 5.36588369481349339860294761879, 6.10486610480185424206658058416, 6.36198654373467615234893955818, 6.61926369665901221466870868365, 7.08435344514735874699303383882, 7.51577421793174998945463067678, 8.139743426164762426896802064442, 8.820569452891575312035096497533