L(s) = 1 | − 2·4-s + 9-s − 14·11-s + 3·16-s − 2·19-s − 4·29-s − 10·31-s − 2·36-s − 18·41-s + 28·44-s + 17·49-s + 28·59-s + 10·61-s − 4·64-s − 58·71-s + 4·76-s + 12·81-s − 14·99-s − 4·101-s − 42·109-s + 8·116-s + 85·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 4-s + 1/3·9-s − 4.22·11-s + 3/4·16-s − 0.458·19-s − 0.742·29-s − 1.79·31-s − 1/3·36-s − 2.81·41-s + 4.22·44-s + 17/7·49-s + 3.64·59-s + 1.28·61-s − 1/2·64-s − 6.88·71-s + 0.458·76-s + 4/3·81-s − 1.40·99-s − 0.398·101-s − 4.02·109-s + 0.742·116-s + 7.72·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2547357215\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2547357215\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 3 | $C_4\times C_2$ | \( 1 - T^{2} - 11 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 17 T^{2} + 141 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 729 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 321 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 10766 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 1026 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 20054 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 341 T^{2} + 47625 T^{4} - 341 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.05069534485786093392430640588, −6.84461407946393813270972277500, −6.73309595043211026452635701941, −6.39312512637828103682891808541, −5.74773463783686144049757919619, −5.66066747521837823662955915601, −5.62232521436253832620673770507, −5.47567020337299612441829422783, −5.35727916230254136361379841954, −5.04265455870099342961153824576, −4.66665806121396910012215287151, −4.56887612037621709218770016165, −4.43874567944835561252862202381, −3.81331076608123693239281415861, −3.69966725540277802712283340580, −3.68607717001711619305555499634, −3.18433928996830021773184473887, −2.72714639332752488043557049530, −2.58828813060652595344920831582, −2.53431098249378248139891989219, −2.10826978460110325798968108821, −1.59339858275510356301425609484, −1.43083537192613349746032026409, −0.51602776230605901476758683100, −0.18563452770645763297984841007,
0.18563452770645763297984841007, 0.51602776230605901476758683100, 1.43083537192613349746032026409, 1.59339858275510356301425609484, 2.10826978460110325798968108821, 2.53431098249378248139891989219, 2.58828813060652595344920831582, 2.72714639332752488043557049530, 3.18433928996830021773184473887, 3.68607717001711619305555499634, 3.69966725540277802712283340580, 3.81331076608123693239281415861, 4.43874567944835561252862202381, 4.56887612037621709218770016165, 4.66665806121396910012215287151, 5.04265455870099342961153824576, 5.35727916230254136361379841954, 5.47567020337299612441829422783, 5.62232521436253832620673770507, 5.66066747521837823662955915601, 5.74773463783686144049757919619, 6.39312512637828103682891808541, 6.73309595043211026452635701941, 6.84461407946393813270972277500, 7.05069534485786093392430640588