| L(s) = 1 | − 4·2-s + 8·4-s + 4·7-s − 8·8-s + 16·11-s − 16·14-s − 4·16-s − 64·22-s + 32·28-s + 32·32-s + 128·44-s + 8·49-s − 32·56-s − 64·64-s − 28·73-s + 64·77-s − 9·81-s + 16·83-s − 128·88-s + 4·97-s − 32·98-s + 56·101-s − 28·103-s + 32·107-s − 16·112-s + 128·121-s + 127-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s + 1.51·7-s − 2.82·8-s + 4.82·11-s − 4.27·14-s − 16-s − 13.6·22-s + 6.04·28-s + 5.65·32-s + 19.2·44-s + 8/7·49-s − 4.27·56-s − 8·64-s − 3.27·73-s + 7.29·77-s − 81-s + 1.75·83-s − 13.6·88-s + 0.406·97-s − 3.23·98-s + 5.57·101-s − 2.75·103-s + 3.09·107-s − 1.51·112-s + 11.6·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327318890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327318890\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 10 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 190 T^{2} + p^{2} T^{4} ) \) |
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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
−7.82665556488751458369033807556, −7.47226484234625763329023682085, −7.46371395680884475988775227647, −7.04057098920862255678748749343, −6.97185758403421616224407450243, −6.60409095948804245191051268477, −6.30748092651142157152460611905, −6.27071516146700591994077048588, −6.17844439028900741561326201202, −5.63044617403558649956522369252, −5.12390441831801024183499735027, −5.10209585688561658187757202988, −4.45811430851062620234667417112, −4.33095519868326383186402466556, −4.24663673862350298665058198454, −4.08994583071541737417085566521, −3.50785130468699690373152310379, −3.30460199406621690583786400676, −2.85433170204208892074268218745, −2.12745208507018920783045265597, −1.85002080543130363494218525620, −1.69943688689389888540800795452, −1.44150347672323802627405578243, −0.906517894310570304525225317506, −0.792171014388010220229700700432,
0.792171014388010220229700700432, 0.906517894310570304525225317506, 1.44150347672323802627405578243, 1.69943688689389888540800795452, 1.85002080543130363494218525620, 2.12745208507018920783045265597, 2.85433170204208892074268218745, 3.30460199406621690583786400676, 3.50785130468699690373152310379, 4.08994583071541737417085566521, 4.24663673862350298665058198454, 4.33095519868326383186402466556, 4.45811430851062620234667417112, 5.10209585688561658187757202988, 5.12390441831801024183499735027, 5.63044617403558649956522369252, 6.17844439028900741561326201202, 6.27071516146700591994077048588, 6.30748092651142157152460611905, 6.60409095948804245191051268477, 6.97185758403421616224407450243, 7.04057098920862255678748749343, 7.46371395680884475988775227647, 7.47226484234625763329023682085, 7.82665556488751458369033807556
