| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 4·6-s + 2·8-s + 9-s − 6·11-s + 2·12-s − 6·13-s − 4·16-s − 8·17-s − 2·18-s + 12·22-s + 2·23-s + 4·24-s + 12·26-s − 2·27-s + 4·29-s + 24·31-s + 2·32-s − 12·33-s + 16·34-s + 36-s − 2·37-s − 12·39-s − 4·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.63·6-s + 0.707·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 1.66·13-s − 16-s − 1.94·17-s − 0.471·18-s + 2.55·22-s + 0.417·23-s + 0.816·24-s + 2.35·26-s − 0.384·27-s + 0.742·29-s + 4.31·31-s + 0.353·32-s − 2.08·33-s + 2.74·34-s + 1/6·36-s − 0.328·37-s − 1.92·39-s − 0.609·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.748308312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748308312\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 - 4 T^{2} - 33 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 24 T^{2} + 48 T^{3} + 223 T^{4} + 48 p T^{5} + 24 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 28 T^{2} + 423 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 33 T^{2} + 18 T^{3} + 748 T^{4} + 18 p T^{5} - 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 6 T^{2} + 144 T^{3} - 821 T^{4} + 144 p T^{5} - 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 88 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T + 19 T^{2} - 178 T^{3} - 1292 T^{4} - 178 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 8 T^{2} - 1617 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 64 T^{2} - 24 T^{3} + 3863 T^{4} - 24 p T^{5} - 64 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 10 T + 43 T^{2} - 650 T^{3} - 6572 T^{4} - 650 p T^{5} + 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 22 T + 231 T^{2} - 2442 T^{3} + 24604 T^{4} - 2442 p T^{5} + 231 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 131 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 120 T^{2} - 48 T^{3} + 20239 T^{4} - 48 p T^{5} - 120 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 22 T + 209 T^{2} + 1782 T^{3} + 19268 T^{4} + 1782 p T^{5} + 209 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.58685710275031008715913449943, −6.49923626064981647931794752368, −6.11888003150890001617134066072, −6.03125250688740909450100964159, −5.68377017597842901947514923931, −5.52024240317278804709362307563, −4.98022081490446578002071154662, −4.95598383777951007707517980665, −4.90114992106952499241106154258, −4.64871640336675373919830711312, −4.43055056545650869758259520738, −4.25149522225860850612567155033, −3.73176243748728570130513870764, −3.63940060815142969651709250651, −3.45963764086113994637062361783, −2.96214831645269989493432821549, −2.63053037129713109333971092588, −2.35334869705212291475025570743, −2.30632685065632745382054816566, −2.30580182948428761355543727325, −2.28055618994058718114731061720, −1.20454305175708519780431294111, −1.08878091351674471136465092490, −0.52381295905200796235195739110, −0.48137090060201410843295615576,
0.48137090060201410843295615576, 0.52381295905200796235195739110, 1.08878091351674471136465092490, 1.20454305175708519780431294111, 2.28055618994058718114731061720, 2.30580182948428761355543727325, 2.30632685065632745382054816566, 2.35334869705212291475025570743, 2.63053037129713109333971092588, 2.96214831645269989493432821549, 3.45963764086113994637062361783, 3.63940060815142969651709250651, 3.73176243748728570130513870764, 4.25149522225860850612567155033, 4.43055056545650869758259520738, 4.64871640336675373919830711312, 4.90114992106952499241106154258, 4.95598383777951007707517980665, 4.98022081490446578002071154662, 5.52024240317278804709362307563, 5.68377017597842901947514923931, 6.03125250688740909450100964159, 6.11888003150890001617134066072, 6.49923626064981647931794752368, 6.58685710275031008715913449943
