| L(s) = 1 | − 12·7-s + 4·9-s − 12·23-s − 24·31-s + 24·41-s + 12·47-s + 68·49-s − 48·63-s + 24·71-s + 16·73-s − 48·79-s + 6·81-s − 24·89-s − 16·97-s − 12·103-s + 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 144·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 4.53·7-s + 4/3·9-s − 2.50·23-s − 4.31·31-s + 3.74·41-s + 1.75·47-s + 68/7·49-s − 6.04·63-s + 2.84·71-s + 1.87·73-s − 5.40·79-s + 2/3·81-s − 2.54·89-s − 1.62·97-s − 1.18·103-s + 2.25·113-s + 1.81·121-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 + 11.3·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7372505403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7372505403\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13002 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71232668698640180797667246562, −6.54982041597820559567011241123, −6.42063915075709875881832336038, −5.89347622653997201109091335189, −5.74594863772436518834234912207, −5.73933794685541887185471948824, −5.72495468034720360008856466359, −5.49531226044552528410680903260, −5.05734636538964618529694229974, −4.36156406636242263863830466630, −4.36142487485072428769078152476, −4.07990890101526082200394947437, −3.99733779439817723181855403385, −3.95983321760789105428529623148, −3.38813411889109783797131683705, −3.32242106923152186483872695255, −3.13632651833736111570493731919, −2.92334236876681849653032330549, −2.36071409947482893284312822335, −2.30860160139900247381029996398, −2.01353434820075179885798404821, −1.60447023758483719775756512531, −1.08704178345478849237219758118, −0.41373076572827071569836634462, −0.34267446449125540644934033319,
0.34267446449125540644934033319, 0.41373076572827071569836634462, 1.08704178345478849237219758118, 1.60447023758483719775756512531, 2.01353434820075179885798404821, 2.30860160139900247381029996398, 2.36071409947482893284312822335, 2.92334236876681849653032330549, 3.13632651833736111570493731919, 3.32242106923152186483872695255, 3.38813411889109783797131683705, 3.95983321760789105428529623148, 3.99733779439817723181855403385, 4.07990890101526082200394947437, 4.36142487485072428769078152476, 4.36156406636242263863830466630, 5.05734636538964618529694229974, 5.49531226044552528410680903260, 5.72495468034720360008856466359, 5.73933794685541887185471948824, 5.74594863772436518834234912207, 5.89347622653997201109091335189, 6.42063915075709875881832336038, 6.54982041597820559567011241123, 6.71232668698640180797667246562
