| L(s) = 1 | − 5·9-s − 6·11-s + 16·19-s + 2·31-s + 12·41-s + 11·49-s − 6·59-s − 16·61-s + 36·71-s − 38·79-s + 9·81-s − 48·89-s + 30·99-s + 28·109-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 80·171-s + 173-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 1.80·11-s + 3.67·19-s + 0.359·31-s + 1.87·41-s + 11/7·49-s − 0.781·59-s − 2.04·61-s + 4.27·71-s − 4.27·79-s + 81-s − 5.08·89-s + 3.01·99-s + 2.68·109-s − 0.454·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s − 6.11·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6454833905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6454833905\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 11 T^{2} + 120 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 47 T^{2} + 1056 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 71 T^{2} + 2244 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 71 T^{2} + 3564 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 59 T^{2} + 3432 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 2235 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 3 T + 112 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 119 T^{2} + 12444 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 18 T + 190 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 23070 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 19 T + 240 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 95 T^{2} + 12396 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 14970 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.08537466247685279436979678752, −6.54653118790060949191243870281, −6.53931415872894783196406638889, −6.14968604949538970041210224018, −5.78885935799973599979652706133, −5.69432107586897634778012869017, −5.62659194011390374015187593192, −5.31322163217149230670419827598, −5.27153206453287786952685734901, −5.14446684672611629003449466242, −4.54146561820402092931478523090, −4.42520596518695425892677949688, −4.32542845598399977158405718593, −3.86713782187989022841320654438, −3.51614212176946648121014132556, −3.22697954595479369922417062488, −3.02845847607244720441885845080, −2.92358654171637993961216996035, −2.67809936742868700766156276768, −2.37466840962880830402842116976, −2.16796698158776734232506981521, −1.48810797752491127058424229651, −1.12660302038682898917331826854, −0.934878594472223597544407425988, −0.18410692662607783722903483878,
0.18410692662607783722903483878, 0.934878594472223597544407425988, 1.12660302038682898917331826854, 1.48810797752491127058424229651, 2.16796698158776734232506981521, 2.37466840962880830402842116976, 2.67809936742868700766156276768, 2.92358654171637993961216996035, 3.02845847607244720441885845080, 3.22697954595479369922417062488, 3.51614212176946648121014132556, 3.86713782187989022841320654438, 4.32542845598399977158405718593, 4.42520596518695425892677949688, 4.54146561820402092931478523090, 5.14446684672611629003449466242, 5.27153206453287786952685734901, 5.31322163217149230670419827598, 5.62659194011390374015187593192, 5.69432107586897634778012869017, 5.78885935799973599979652706133, 6.14968604949538970041210224018, 6.53931415872894783196406638889, 6.54653118790060949191243870281, 7.08537466247685279436979678752
