| L(s) = 1 | − 4-s + 6·7-s + 6·11-s + 16-s + 6·25-s − 6·28-s − 12·37-s − 4·41-s − 6·44-s − 16·47-s + 13·49-s + 22·53-s − 64-s + 16·67-s − 4·71-s − 2·73-s + 36·77-s + 2·83-s − 6·100-s + 36·101-s − 26·107-s + 6·112-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.26·7-s + 1.80·11-s + 1/4·16-s + 6/5·25-s − 1.13·28-s − 1.97·37-s − 0.624·41-s − 0.904·44-s − 2.33·47-s + 13/7·49-s + 3.02·53-s − 1/8·64-s + 1.95·67-s − 0.474·71-s − 0.234·73-s + 4.10·77-s + 0.219·83-s − 3/5·100-s + 3.58·101-s − 2.51·107-s + 0.566·112-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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
| \(L(1)\) |
\(\approx\) |
\(2.652901792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.652901792\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.69400881358725267256457404650, −10.35324407771333599938640765703, −9.944649559517724197208680586103, −9.270884385815959776012469963259, −8.795006404857291260588393017258, −8.736775375156564642099615121598, −8.139896521062728705270584459380, −7.914048780608247531027610480318, −7.14032881057699665746524126730, −6.78032360389036728465296205528, −6.45875459614170990089478952886, −5.42367013670253507092974310989, −5.33618170766450893144817516761, −4.71083335473602973827068491188, −4.38768840052886046336718122271, −3.74255874557964379480125673889, −3.28590846214964201912276330378, −2.12805965352942002407918436102, −1.61265117025980691010863586818, −1.01765075521813034107061591351,
1.01765075521813034107061591351, 1.61265117025980691010863586818, 2.12805965352942002407918436102, 3.28590846214964201912276330378, 3.74255874557964379480125673889, 4.38768840052886046336718122271, 4.71083335473602973827068491188, 5.33618170766450893144817516761, 5.42367013670253507092974310989, 6.45875459614170990089478952886, 6.78032360389036728465296205528, 7.14032881057699665746524126730, 7.914048780608247531027610480318, 8.139896521062728705270584459380, 8.736775375156564642099615121598, 8.795006404857291260588393017258, 9.270884385815959776012469963259, 9.944649559517724197208680586103, 10.35324407771333599938640765703, 10.69400881358725267256457404650
