L(s) = 1 | − 4·5-s + 4·19-s + 2·25-s − 8·29-s + 24·31-s + 32·41-s − 12·49-s − 24·59-s + 24·71-s − 8·79-s + 2·81-s − 8·89-s − 16·95-s + 16·101-s − 44·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s − 96·155-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 1.78·5-s + 0.917·19-s + 2/5·25-s − 1.48·29-s + 4.31·31-s + 4.99·41-s − 1.71·49-s − 3.12·59-s + 2.84·71-s − 0.900·79-s + 2/9·81-s − 0.847·89-s − 1.64·95-s + 1.59·101-s − 4·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s − 7.71·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.492601044\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.492601044\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 2158 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 602 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 4398 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 25342 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25222 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4758 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 40 T^{2} - 5282 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.75748918684056387131139544478, −6.42881252601734286860390947471, −6.38917974441952988610920276054, −6.11796060274431820961882639778, −5.85924386847324735692133598094, −5.78904643638252697454710700331, −5.58091973879745802656206820661, −4.98754678882548136076249449768, −4.83756397347429447969902814539, −4.71061199632861557512404499118, −4.70923249489100859106265713991, −4.08466713788181593607442808809, −4.04581059692740183685151766327, −3.92019161124086504698463518031, −3.77424793644705896764644534687, −3.17061122152241894742194018719, −3.10673657472407400143653004563, −2.80689279243106968751623605986, −2.57464255619718665494147186818, −2.41357424528843728019050976638, −1.82542113696614157635501596830, −1.47694462615177501787957281178, −1.12207663111506141448434994609, −0.55281636755501439907646968997, −0.54003496349931498685870966663,
0.54003496349931498685870966663, 0.55281636755501439907646968997, 1.12207663111506141448434994609, 1.47694462615177501787957281178, 1.82542113696614157635501596830, 2.41357424528843728019050976638, 2.57464255619718665494147186818, 2.80689279243106968751623605986, 3.10673657472407400143653004563, 3.17061122152241894742194018719, 3.77424793644705896764644534687, 3.92019161124086504698463518031, 4.04581059692740183685151766327, 4.08466713788181593607442808809, 4.70923249489100859106265713991, 4.71061199632861557512404499118, 4.83756397347429447969902814539, 4.98754678882548136076249449768, 5.58091973879745802656206820661, 5.78904643638252697454710700331, 5.85924386847324735692133598094, 6.11796060274431820961882639778, 6.38917974441952988610920276054, 6.42881252601734286860390947471, 6.75748918684056387131139544478