| L(s) = 1 | + 64·4-s − 972·9-s + 3.07e3·16-s − 6.22e4·36-s + 1.60e5·53-s + 1.31e5·64-s + 5.90e5·81-s − 9.76e5·113-s + 6.44e5·121-s + 127-s + 131-s + 137-s + 139-s − 2.98e6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4·9-s + 3·16-s − 8·36-s + 7.87·53-s + 4·64-s + 10·81-s − 7.19·113-s + 4·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 12·144-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.208667171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.208667171\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 5 | $C_4\times C_2$ | \( 1 + 2953968 T^{4} + p^{20} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 13 | $C_4\times C_2$ | \( 1 + 198997054800 T^{4} + p^{20} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 3961557501600 T^{4} + p^{20} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 25263800 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 138236664 T^{2} + p^{10} T^{4} )^{2} \) |
| 41 | $C_4\times C_2$ | \( 1 + 21494812402951200 T^{4} + p^{20} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - 40244 T + p^{5} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 61 | $C_4\times C_2$ | \( 1 - 1384975738022989200 T^{4} + p^{20} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 73 | $C_4\times C_2$ | \( 1 - 6691164549087796800 T^{4} + p^{20} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 89 | $C_4\times C_2$ | \( 1 - 61395541391328705600 T^{4} + p^{20} T^{8} \) |
| 97 | $C_4\times C_2$ | \( 1 - \)\(12\!\cdots\!00\)\( T^{4} + p^{20} T^{8} \) |
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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
−8.188958397029739237265709584792, −8.145409300583875989431934579921, −7.52108006936724189076724509933, −7.50908648862349524488078764084, −7.10761735770395950935888897596, −6.61787686429467956664282313811, −6.56190581843481974470292827113, −6.47885014032101398394889496473, −5.67311404059736403550945721059, −5.61865759389355428060384210100, −5.60708287255830863203162220845, −5.47476871664240343811304867410, −5.14386284026118418054052337735, −4.19673717931237775843734201866, −4.08791126780644038369862432391, −3.64324893244988672462111548198, −3.10779360888616644925267813702, −3.02756886357468617885944325999, −2.71582671970456079147103676174, −2.43080507255299197544509939296, −2.07414249747338281744908524688, −1.88980882934247760277662114248, −0.928841662122841478053859936659, −0.64488758142805395885467181468, −0.46358899821113702898255179132,
0.46358899821113702898255179132, 0.64488758142805395885467181468, 0.928841662122841478053859936659, 1.88980882934247760277662114248, 2.07414249747338281744908524688, 2.43080507255299197544509939296, 2.71582671970456079147103676174, 3.02756886357468617885944325999, 3.10779360888616644925267813702, 3.64324893244988672462111548198, 4.08791126780644038369862432391, 4.19673717931237775843734201866, 5.14386284026118418054052337735, 5.47476871664240343811304867410, 5.60708287255830863203162220845, 5.61865759389355428060384210100, 5.67311404059736403550945721059, 6.47885014032101398394889496473, 6.56190581843481974470292827113, 6.61787686429467956664282313811, 7.10761735770395950935888897596, 7.50908648862349524488078764084, 7.52108006936724189076724509933, 8.145409300583875989431934579921, 8.188958397029739237265709584792
