| L(s) = 1 | − 8·5-s − 2·9-s + 84·17-s − 34·25-s + 48·29-s − 8·37-s − 32·41-s + 16·45-s − 34·49-s − 40·53-s − 64·61-s + 208·73-s + 49·81-s − 672·85-s − 352·89-s − 472·97-s + 80·101-s + 56·109-s + 56·113-s + 332·121-s + 528·125-s + 127-s + 131-s + 137-s + 139-s − 384·145-s + 149-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 2/9·9-s + 4.94·17-s − 1.35·25-s + 1.65·29-s − 0.216·37-s − 0.780·41-s + 0.355·45-s − 0.693·49-s − 0.754·53-s − 1.04·61-s + 2.84·73-s + 0.604·81-s − 7.90·85-s − 3.95·89-s − 4.86·97-s + 0.792·101-s + 0.513·109-s + 0.495·113-s + 2.74·121-s + 4.22·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.64·145-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.790147339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.790147339\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 3 | $D_{4}$$\times$$D_{4}$ | \( ( 1 - 4 T + p^{2} T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )( 1 + 4 T + p^{2} T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 5 | $D_{4}$ | \( ( 1 + 4 T + 41 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 34 T^{2} + 4259 T^{4} + 34 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 332 T^{2} + 53510 T^{4} - 332 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 21 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 364 T^{2} + 24198 T^{4} - 364 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1268 T^{2} + 821030 T^{4} - 1268 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 24 T + 1358 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1412 T^{2} + 1493510 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 1689 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 16 T + 98 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 3886 T^{2} + 7765539 T^{4} - 3886 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 274 T^{2} + 9637523 T^{4} + 274 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 20 T + 4418 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 164 p T^{2} + 43327878 T^{4} - 164 p^{5} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 32 T + 6866 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 14572 T^{2} + 92309766 T^{4} - 14572 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 15134 T^{2} + 106881443 T^{4} - 15134 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 104 T + 9150 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 5684 T^{2} + 65469158 T^{4} - 5684 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 26260 T^{2} + 267246150 T^{4} - 26260 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 176 T + 21038 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 236 T + 27542 T^{2} + 236 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.15522794752902748767630561643, −7.14120948891458540015487316174, −6.54965721865656015388793012537, −6.53589119839647468450279366096, −6.17664655738859106786292590589, −5.87018462593299127664994200344, −5.69795919049884456658615334713, −5.49183175496949652932823065498, −5.31280678028012520490396732834, −5.04093999951866267038717424175, −4.80254673224194781424153517583, −4.39941217358321235486249047647, −4.15367380499473365964483031410, −3.80487398866009136006525505782, −3.79321338677471760427486714049, −3.42424844202454944310720482820, −3.14558843641988701394773690327, −2.96746127057233078744886705123, −2.86280551426022797477042193992, −2.21025785428274628577873873401, −1.73785585733562199840986072094, −1.34823875054861798318922661581, −1.24262007365230265857488816521, −0.57234663625823645418018803495, −0.38793990955894626315804131272,
0.38793990955894626315804131272, 0.57234663625823645418018803495, 1.24262007365230265857488816521, 1.34823875054861798318922661581, 1.73785585733562199840986072094, 2.21025785428274628577873873401, 2.86280551426022797477042193992, 2.96746127057233078744886705123, 3.14558843641988701394773690327, 3.42424844202454944310720482820, 3.79321338677471760427486714049, 3.80487398866009136006525505782, 4.15367380499473365964483031410, 4.39941217358321235486249047647, 4.80254673224194781424153517583, 5.04093999951866267038717424175, 5.31280678028012520490396732834, 5.49183175496949652932823065498, 5.69795919049884456658615334713, 5.87018462593299127664994200344, 6.17664655738859106786292590589, 6.53589119839647468450279366096, 6.54965721865656015388793012537, 7.14120948891458540015487316174, 7.15522794752902748767630561643
