| L(s) = 1 | + 8·7-s + 4·9-s + 4·25-s − 32·31-s + 34·49-s + 32·63-s − 6·81-s − 16·103-s − 72·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 4/3·9-s + 4/5·25-s − 5.74·31-s + 34/7·49-s + 4.03·63-s − 2/3·81-s − 1.57·103-s − 6.77·113-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 + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.974753264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.974753264\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−6.71696830135905326800491183742, −6.69008259656637490972590677662, −5.88167872646949444854727333832, −5.84598974630436377765673290252, −5.76040004110084872342578564834, −5.46332733702856995997139578999, −5.31390039354101702332540263653, −5.11870142618711833205801656335, −4.77484478179335369302540123919, −4.68752005536493134425437009207, −4.63469641277580437120517900699, −4.12331770300204152270433532727, −3.96038811704835909635138117478, −3.75542677648097625614275171449, −3.65158592882777264273271355498, −3.47310751387342768724261869625, −2.65584026475610212031884129966, −2.62292988788677763959578900250, −2.50588892202341957149153976129, −1.83957233770709877289524295112, −1.61910428636712216156771060209, −1.60488561404812693188391975590, −1.48779104086079354878900818507, −1.01000571017339276369949623800, −0.26578833208435336949568569617,
0.26578833208435336949568569617, 1.01000571017339276369949623800, 1.48779104086079354878900818507, 1.60488561404812693188391975590, 1.61910428636712216156771060209, 1.83957233770709877289524295112, 2.50588892202341957149153976129, 2.62292988788677763959578900250, 2.65584026475610212031884129966, 3.47310751387342768724261869625, 3.65158592882777264273271355498, 3.75542677648097625614275171449, 3.96038811704835909635138117478, 4.12331770300204152270433532727, 4.63469641277580437120517900699, 4.68752005536493134425437009207, 4.77484478179335369302540123919, 5.11870142618711833205801656335, 5.31390039354101702332540263653, 5.46332733702856995997139578999, 5.76040004110084872342578564834, 5.84598974630436377765673290252, 5.88167872646949444854727333832, 6.69008259656637490972590677662, 6.71696830135905326800491183742
