| L(s) = 1 | + 2-s + 4-s + 8-s + 3·11-s + 16-s + 3·17-s − 2·19-s + 3·22-s − 25-s + 32-s + 3·34-s − 2·38-s + 9·41-s + 7·43-s + 3·44-s − 10·49-s − 50-s + 15·59-s + 64-s + 10·67-s + 3·68-s − 14·73-s − 2·76-s + 9·82-s − 21·83-s + 7·86-s + 3·88-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.904·11-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.639·22-s − 1/5·25-s + 0.176·32-s + 0.514·34-s − 0.324·38-s + 1.40·41-s + 1.06·43-s + 0.452·44-s − 1.42·49-s − 0.141·50-s + 1.95·59-s + 1/8·64-s + 1.22·67-s + 0.363·68-s − 1.63·73-s − 0.229·76-s + 0.993·82-s − 2.30·83-s + 0.754·86-s + 0.319·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.530757931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.530757931\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.800806926711803704531501153338, −9.033385003825437670452219591228, −8.737305069365379630244767686215, −8.011587325014383079871280078995, −7.58801805180469460627655375313, −7.00757760791682427982478518741, −6.52325437143589623787770647758, −5.94040724255504973565807559548, −5.56092558365466387252393932257, −4.81928813089289256947751730786, −4.18121972591858360741511286021, −3.77143197463082969899768658530, −2.98752071666973071322683578677, −2.21617641911960587908713462526, −1.19397967770804096946527711593,
1.19397967770804096946527711593, 2.21617641911960587908713462526, 2.98752071666973071322683578677, 3.77143197463082969899768658530, 4.18121972591858360741511286021, 4.81928813089289256947751730786, 5.56092558365466387252393932257, 5.94040724255504973565807559548, 6.52325437143589623787770647758, 7.00757760791682427982478518741, 7.58801805180469460627655375313, 8.011587325014383079871280078995, 8.737305069365379630244767686215, 9.033385003825437670452219591228, 9.800806926711803704531501153338
