| L(s) = 1 | − 5·9-s + 6·11-s − 4·23-s − 14·29-s − 4·37-s − 16·43-s − 20·53-s − 32·67-s + 4·71-s + 6·79-s + 11·81-s − 30·99-s + 32·107-s + 30·109-s + 4·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 45·169-s + 173-s + ⋯ |
| L(s) = 1 | − 5/3·9-s + 1.80·11-s − 0.834·23-s − 2.59·29-s − 0.657·37-s − 2.43·43-s − 2.74·53-s − 3.90·67-s + 0.474·71-s + 0.675·79-s + 11/9·81-s − 3.01·99-s + 3.09·107-s + 2.87·109-s + 0.376·113-s − 0.0909·121-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 + 0.0783·163-s + 0.0773·167-s − 3.46·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 + 5 T^{2} + 14 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 45 T^{2} + 834 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 55 T^{2} + 1324 T^{4} + 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 18 T^{2} + 762 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 68 T^{2} + 2422 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 108 T^{2} + 5622 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 7 T^{2} + 3928 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 2 T^{2} + 3642 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 138 T^{2} + 10194 T^{4} + 138 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 71 | $D_{4}$ | \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 8 T^{2} + 10510 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 3 T + 68 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 226 T^{2} + 24538 T^{4} + 226 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 108 T^{2} + 16134 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 287 T^{2} + 38908 T^{4} + 287 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.93959479188296081057633413294, −5.50575548717016983236106656827, −5.26328480177210394748135250394, −5.20657228963422596031769159579, −5.12248906872962437479195960691, −4.81442471937369900119739657895, −4.64861364549522201860331008376, −4.62340999389892942432564292119, −4.20624155425914849012546182307, −3.89008540063509879921097488346, −3.81754523276686198788065713942, −3.70911501117371450589902050365, −3.70814588695238193514486259255, −3.29492606294194192277514378393, −3.06680414391097385391643323548, −3.00879625419353515426077597876, −2.88401880860538643665361771132, −2.41652723016144481849923341750, −2.27283999453737262827712080710, −1.97626449736627437579447795950, −1.79605028772165359154786368598, −1.74318184197052990340860898513, −1.20771756164731265725654856295, −1.15882144866024174080270011825, −1.09318639742392861855754772909, 0, 0, 0, 0,
1.09318639742392861855754772909, 1.15882144866024174080270011825, 1.20771756164731265725654856295, 1.74318184197052990340860898513, 1.79605028772165359154786368598, 1.97626449736627437579447795950, 2.27283999453737262827712080710, 2.41652723016144481849923341750, 2.88401880860538643665361771132, 3.00879625419353515426077597876, 3.06680414391097385391643323548, 3.29492606294194192277514378393, 3.70814588695238193514486259255, 3.70911501117371450589902050365, 3.81754523276686198788065713942, 3.89008540063509879921097488346, 4.20624155425914849012546182307, 4.62340999389892942432564292119, 4.64861364549522201860331008376, 4.81442471937369900119739657895, 5.12248906872962437479195960691, 5.20657228963422596031769159579, 5.26328480177210394748135250394, 5.50575548717016983236106656827, 5.93959479188296081057633413294
