| L(s) = 1 | + 4-s − 2·9-s − 4·11-s + 4·16-s + 4·19-s + 8·29-s − 2·36-s − 32·41-s − 4·44-s − 2·49-s + 12·59-s − 8·61-s + 11·64-s − 20·71-s + 4·76-s − 16·79-s + 3·81-s − 4·89-s + 8·99-s − 44·101-s − 28·109-s + 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/3·9-s − 1.20·11-s + 16-s + 0.917·19-s + 1.48·29-s − 1/3·36-s − 4.99·41-s − 0.603·44-s − 2/7·49-s + 1.56·59-s − 1.02·61-s + 11/8·64-s − 2.37·71-s + 0.458·76-s − 1.80·79-s + 1/3·81-s − 0.423·89-s + 0.804·99-s − 4.37·101-s − 2.68·109-s + 0.742·116-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.173044024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.173044024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 24 T^{2} + 542 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 5467 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 90 T^{2} + 5003 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_4$ | \( ( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 146 T^{2} + 11427 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 10 T + 147 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 30238 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 15382 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.82779121250595269589007569860, −7.70529078176513594536598361853, −7.33474140706265203583111947204, −7.02135852238868284906466194689, −7.01504206603295341141920793164, −6.53698472330534938503928165471, −6.38209003437364867952162510224, −6.37372105154529693582361468244, −5.76125464253528751313768579938, −5.51662501056955730919185238440, −5.30313492292577676170906596520, −5.25222771584494571758207954451, −4.96767850459350227327135990783, −4.75178273847731043647911131995, −4.08066378282328285310519425593, −3.91637536216849347108536082304, −3.83966297596316986770140245596, −2.97834451304383027415082022637, −2.96822422112917751123288128406, −2.95626352174807229827986193188, −2.66502739460181134897732335368, −1.79005906929978794111508077973, −1.68028554892747885178977757722, −1.27679661244757547946785259237, −0.34176444194724140528034704113,
0.34176444194724140528034704113, 1.27679661244757547946785259237, 1.68028554892747885178977757722, 1.79005906929978794111508077973, 2.66502739460181134897732335368, 2.95626352174807229827986193188, 2.96822422112917751123288128406, 2.97834451304383027415082022637, 3.83966297596316986770140245596, 3.91637536216849347108536082304, 4.08066378282328285310519425593, 4.75178273847731043647911131995, 4.96767850459350227327135990783, 5.25222771584494571758207954451, 5.30313492292577676170906596520, 5.51662501056955730919185238440, 5.76125464253528751313768579938, 6.37372105154529693582361468244, 6.38209003437364867952162510224, 6.53698472330534938503928165471, 7.01504206603295341141920793164, 7.02135852238868284906466194689, 7.33474140706265203583111947204, 7.70529078176513594536598361853, 7.82779121250595269589007569860
