| L(s) = 1 | + 2-s + 4-s + 8-s + 10·13-s + 16-s + 6·17-s − 10·25-s + 10·26-s − 6·29-s + 32-s + 6·34-s + 4·37-s − 12·41-s + 49-s − 10·50-s + 10·52-s + 6·53-s − 6·58-s − 20·61-s + 64-s + 6·68-s + 4·73-s + 4·74-s − 12·82-s + 30·89-s + 16·97-s + 98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2.77·13-s + 1/4·16-s + 1.45·17-s − 2·25-s + 1.96·26-s − 1.11·29-s + 0.176·32-s + 1.02·34-s + 0.657·37-s − 1.87·41-s + 1/7·49-s − 1.41·50-s + 1.38·52-s + 0.824·53-s − 0.787·58-s − 2.56·61-s + 1/8·64-s + 0.727·68-s + 0.468·73-s + 0.464·74-s − 1.32·82-s + 3.17·89-s + 1.62·97-s + 0.101·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143072 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143072 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.071419677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.071419677\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.949595272551558511484925825699, −7.66081859530557802153309309893, −7.24729027002187867605972756035, −6.38983991712328031863989081821, −6.29670817383847541122613475379, −5.75498451669473966849176010933, −5.61879984028395623126782300752, −4.92409316048517899781293445732, −4.30645127647783484429164317499, −3.77842535731702409100199368470, −3.34773629764867655679280376645, −3.29476397189543990973285365054, −2.01879358291610012679245055865, −1.68869994047015348511995782895, −0.856193565353031792325176418947,
0.856193565353031792325176418947, 1.68869994047015348511995782895, 2.01879358291610012679245055865, 3.29476397189543990973285365054, 3.34773629764867655679280376645, 3.77842535731702409100199368470, 4.30645127647783484429164317499, 4.92409316048517899781293445732, 5.61879984028395623126782300752, 5.75498451669473966849176010933, 6.29670817383847541122613475379, 6.38983991712328031863989081821, 7.24729027002187867605972756035, 7.66081859530557802153309309893, 7.949595272551558511484925825699
