| L(s) = 1 | − 4-s + 3·7-s + 4·11-s + 16-s − 8·25-s − 3·28-s + 37-s + 13·41-s − 4·44-s + 6·47-s + 5·49-s − 20·53-s − 64-s + 13·67-s − 7·71-s + 5·73-s + 12·77-s + 16·83-s + 8·100-s + 15·101-s + 18·107-s + 3·112-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.13·7-s + 1.20·11-s + 1/4·16-s − 8/5·25-s − 0.566·28-s + 0.164·37-s + 2.03·41-s − 0.603·44-s + 0.875·47-s + 5/7·49-s − 2.74·53-s − 1/8·64-s + 1.58·67-s − 0.830·71-s + 0.585·73-s + 1.36·77-s + 1.75·83-s + 4/5·100-s + 1.49·101-s + 1.74·107-s + 0.283·112-s − 0.909·121-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
| \(L(1)\) |
\(\approx\) |
\(2.029392846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.029392846\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
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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
−8.617451203359554607219134778477, −8.046872212628095564318428354475, −7.71271300799164520879637406696, −7.47431463174357460930506372091, −6.66820517373916982825524350346, −6.19293401223317918231398682282, −5.86189032764486336106986948203, −5.19187144940836707433831235689, −4.73716492563301369841219066979, −4.18859207178477992353553830929, −3.88149961625522328008614212086, −3.19103623857807470379733923388, −2.23773103016835682004447787097, −1.68912950337368810268678997452, −0.825185096944106396088849162776,
0.825185096944106396088849162776, 1.68912950337368810268678997452, 2.23773103016835682004447787097, 3.19103623857807470379733923388, 3.88149961625522328008614212086, 4.18859207178477992353553830929, 4.73716492563301369841219066979, 5.19187144940836707433831235689, 5.86189032764486336106986948203, 6.19293401223317918231398682282, 6.66820517373916982825524350346, 7.47431463174357460930506372091, 7.71271300799164520879637406696, 8.046872212628095564318428354475, 8.617451203359554607219134778477
