| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 3·9-s − 2·13-s + 5·16-s − 3·17-s + 6·18-s + 12·19-s − 9·25-s − 4·26-s + 6·32-s − 6·34-s + 9·36-s + 24·38-s − 10·43-s + 26·47-s − 13·49-s − 18·50-s − 6·52-s + 24·53-s − 20·59-s + 7·64-s − 4·67-s − 9·68-s + 12·72-s + 36·76-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 9-s − 0.554·13-s + 5/4·16-s − 0.727·17-s + 1.41·18-s + 2.75·19-s − 9/5·25-s − 0.784·26-s + 1.06·32-s − 1.02·34-s + 3/2·36-s + 3.89·38-s − 1.52·43-s + 3.79·47-s − 1.85·49-s − 2.54·50-s − 0.832·52-s + 3.29·53-s − 2.60·59-s + 7/8·64-s − 0.488·67-s − 1.09·68-s + 1.41·72-s + 4.12·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.582540933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.582540933\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.328397420964799854901742898187, −8.601415375369587900969461268055, −7.77469056799720070910967744275, −7.45985560667588965978165906144, −7.23321315314702664404367449530, −6.68766678039395086013377186172, −5.88908450915949561478136129218, −5.68196385649536348701619256830, −5.01781591323975843915294811629, −4.61153453491182141362175201622, −3.94315686955997078455990956255, −3.56289633286702168183431548175, −2.79948963525070016890455292702, −2.11485045552833555294339924745, −1.24541420514313337396809760448,
1.24541420514313337396809760448, 2.11485045552833555294339924745, 2.79948963525070016890455292702, 3.56289633286702168183431548175, 3.94315686955997078455990956255, 4.61153453491182141362175201622, 5.01781591323975843915294811629, 5.68196385649536348701619256830, 5.88908450915949561478136129218, 6.68766678039395086013377186172, 7.23321315314702664404367449530, 7.45985560667588965978165906144, 7.77469056799720070910967744275, 8.601415375369587900969461268055, 9.328397420964799854901742898187
