| L(s) = 1 | − 2·3-s + 3·9-s + 6·13-s + 14·17-s − 2·23-s + 9·25-s − 4·27-s − 2·29-s − 12·39-s + 10·43-s − 49-s − 28·51-s − 12·53-s − 26·61-s + 4·69-s − 18·75-s + 24·79-s + 5·81-s + 4·87-s + 28·101-s + 2·103-s − 4·107-s − 20·113-s + 18·117-s + 13·121-s + 127-s − 20·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.66·13-s + 3.39·17-s − 0.417·23-s + 9/5·25-s − 0.769·27-s − 0.371·29-s − 1.92·39-s + 1.52·43-s − 1/7·49-s − 3.92·51-s − 1.64·53-s − 3.32·61-s + 0.481·69-s − 2.07·75-s + 2.70·79-s + 5/9·81-s + 0.428·87-s + 2.78·101-s + 0.197·103-s − 0.386·107-s − 1.88·113-s + 1.66·117-s + 1.18·121-s + 0.0887·127-s − 1.76·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.859099217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.859099217\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.349767129780511638535051620321, −8.209662568024948702209953936941, −7.67485510987562422449112944898, −7.54664289122515777609877662114, −7.13675936219224482208945724592, −6.59892478696932866614096836323, −6.15677583157293307219287170823, −5.98520933629242487396428346402, −5.74480894316983856240713556968, −5.33017364760809383556233696477, −4.75338849251651190930964375180, −4.69560488741758750890461214685, −3.98620155066313285100279508886, −3.52127693754453556156401590873, −3.19281409100618000644348155891, −2.99781194192239128282687873459, −1.94683708157462223591076479602, −1.46423731735087239600263554104, −0.995126112174070782269970225459, −0.66854931075850297804195872741,
0.66854931075850297804195872741, 0.995126112174070782269970225459, 1.46423731735087239600263554104, 1.94683708157462223591076479602, 2.99781194192239128282687873459, 3.19281409100618000644348155891, 3.52127693754453556156401590873, 3.98620155066313285100279508886, 4.69560488741758750890461214685, 4.75338849251651190930964375180, 5.33017364760809383556233696477, 5.74480894316983856240713556968, 5.98520933629242487396428346402, 6.15677583157293307219287170823, 6.59892478696932866614096836323, 7.13675936219224482208945724592, 7.54664289122515777609877662114, 7.67485510987562422449112944898, 8.209662568024948702209953936941, 8.349767129780511638535051620321
