| L(s) = 1 | + 10·7-s + 5·9-s − 6·17-s + 8·23-s + 9·25-s + 8·31-s − 16·41-s − 18·47-s + 61·49-s + 50·63-s + 14·71-s + 4·73-s − 24·79-s + 16·81-s + 20·89-s − 20·97-s − 16·103-s + 4·113-s − 60·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 30·153-s + ⋯ |
| L(s) = 1 | + 3.77·7-s + 5/3·9-s − 1.45·17-s + 1.66·23-s + 9/5·25-s + 1.43·31-s − 2.49·41-s − 2.62·47-s + 61/7·49-s + 6.29·63-s + 1.66·71-s + 0.468·73-s − 2.70·79-s + 16/9·81-s + 2.11·89-s − 2.03·97-s − 1.57·103-s + 0.376·113-s − 5.50·119-s + 1.63·121-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 − 2.42·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.876662879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.876662879\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.514477519460100319070660023293, −8.335525332932874463740362938349, −8.309923688969863527643966318427, −7.80941370447120824547088487848, −7.15407862067597596101999607691, −7.12850121786104882184048132394, −6.74095158460709907457774286198, −6.41071259876223988354969895620, −5.47096914983727904041618168750, −5.19912225029166627708780619273, −4.73929024737378344585304416958, −4.69728506769153926353137212785, −4.53952015556974212382482756353, −3.97358006233017579008206293703, −3.25406328819077575842572007635, −2.68244049237601477326472104815, −1.90929866465246876289238130698, −1.82247491036870717414203840174, −1.25383794369386559965413548781, −0.930935038927424448153378996885,
0.930935038927424448153378996885, 1.25383794369386559965413548781, 1.82247491036870717414203840174, 1.90929866465246876289238130698, 2.68244049237601477326472104815, 3.25406328819077575842572007635, 3.97358006233017579008206293703, 4.53952015556974212382482756353, 4.69728506769153926353137212785, 4.73929024737378344585304416958, 5.19912225029166627708780619273, 5.47096914983727904041618168750, 6.41071259876223988354969895620, 6.74095158460709907457774286198, 7.12850121786104882184048132394, 7.15407862067597596101999607691, 7.80941370447120824547088487848, 8.309923688969863527643966318427, 8.335525332932874463740362938349, 8.514477519460100319070660023293
