| L(s) = 1 | + 10·9-s − 4·11-s − 8·13-s + 8·19-s + 16·23-s − 8·29-s + 8·31-s + 8·37-s + 8·41-s + 20·43-s − 8·47-s + 16·59-s − 8·61-s + 4·67-s − 24·73-s + 57·81-s − 36·83-s − 24·89-s − 28·97-s − 40·99-s + 8·103-s + 24·109-s + 64·113-s − 80·117-s + 8·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 10/3·9-s − 1.20·11-s − 2.21·13-s + 1.83·19-s + 3.33·23-s − 1.48·29-s + 1.43·31-s + 1.31·37-s + 1.24·41-s + 3.04·43-s − 1.16·47-s + 2.08·59-s − 1.02·61-s + 0.488·67-s − 2.80·73-s + 19/3·81-s − 3.95·83-s − 2.54·89-s − 2.84·97-s − 4.02·99-s + 0.788·103-s + 2.29·109-s + 6.02·113-s − 7.39·117-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.765897760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.765897760\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 17 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 18 T^{2} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 + 14 T^{2} + 467 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 56 T^{3} - 46 T^{4} - 56 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 52 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 152 T^{3} + 578 T^{4} - 152 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 320 T^{3} + 3191 T^{4} - 320 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 232 T^{3} + 1538 T^{4} - 232 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 80 T^{3} - 1169 T^{4} + 80 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 44 T^{3} - 5842 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 8687 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 546 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 36 T + 648 T^{2} + 8100 T^{3} + 81086 T^{4} + 8100 p T^{5} + 648 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 3384 T^{3} + 37058 T^{4} + 3384 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 28 T + 392 T^{2} + 4900 T^{3} + 55166 T^{4} + 4900 p T^{5} + 392 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.24540711618315172626731583655, −7.18484291236185538987941836353, −7.02944431273268137596511653471, −6.93240619363106360047198192706, −6.76154769000676520323967908010, −5.89047693530968064178407972419, −5.78345553161091382466960814130, −5.76902237208018730887263107122, −5.68060530352411107430625892539, −5.01078488177627523989330790361, −4.79003191676737789047365590490, −4.70277011583786354655669338397, −4.56942151507851981626280540177, −4.41091918660200354684190613793, −4.06520034774877815047634847791, −3.76852924375064055021378055185, −3.11570543813214050581105733495, −3.10680322312971677808089268105, −2.86741804465341959484589053762, −2.46213595532823798708603104831, −2.22953969707436585094543426499, −1.76108011685486823386403432172, −1.17888513327873646033439619151, −1.10959431488628920150437446405, −0.63920390808653339447135156787,
0.63920390808653339447135156787, 1.10959431488628920150437446405, 1.17888513327873646033439619151, 1.76108011685486823386403432172, 2.22953969707436585094543426499, 2.46213595532823798708603104831, 2.86741804465341959484589053762, 3.10680322312971677808089268105, 3.11570543813214050581105733495, 3.76852924375064055021378055185, 4.06520034774877815047634847791, 4.41091918660200354684190613793, 4.56942151507851981626280540177, 4.70277011583786354655669338397, 4.79003191676737789047365590490, 5.01078488177627523989330790361, 5.68060530352411107430625892539, 5.76902237208018730887263107122, 5.78345553161091382466960814130, 5.89047693530968064178407972419, 6.76154769000676520323967908010, 6.93240619363106360047198192706, 7.02944431273268137596511653471, 7.18484291236185538987941836353, 7.24540711618315172626731583655
