| L(s) = 1 | + 3·3-s + 8·4-s + 2·7-s + 24·12-s − 22·13-s + 48·16-s + 37·19-s + 6·21-s − 25·25-s − 27·27-s + 16·28-s + 13·31-s + 94·37-s − 66·39-s + 22·43-s + 144·48-s − 45·49-s − 176·52-s + 111·57-s + 121·61-s + 256·64-s + 13·67-s + 46·73-s − 75·75-s + 296·76-s − 11·79-s − 81·81-s + ⋯ |
| L(s) = 1 | + 3-s + 2·4-s + 2/7·7-s + 2·12-s − 1.69·13-s + 3·16-s + 1.94·19-s + 2/7·21-s − 25-s − 27-s + 4/7·28-s + 0.419·31-s + 2.54·37-s − 1.69·39-s + 0.511·43-s + 3·48-s − 0.918·49-s − 3.38·52-s + 1.94·57-s + 1.98·61-s + 4·64-s + 0.194·67-s + 0.630·73-s − 75-s + 3.89·76-s − 0.139·79-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.131346900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.131346900\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 22 T + p^{2} T^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 61 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 - 47 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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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
−11.71713060033849522735567827264, −11.46955939394818327745110718453, −11.25660373742175658794531957095, −10.48829488288820188849849077179, −9.793172155439180332390666150209, −9.748363007805241489698153675972, −9.260934084989844653679725901204, −8.059834272649901480431101846871, −8.051520485740927865966210989715, −7.62947764982484849351923290378, −7.07084190151144290529340672358, −6.72161019973575327186699422367, −5.72469094873573646097481535324, −5.65774866923759257991832112799, −4.73651342509810764751574597705, −3.77009077410526096369120031381, −3.09295236536296196185120877386, −2.53390951046190893298063997490, −2.21733615874305099104968466724, −1.14910784235051958028445678409,
1.14910784235051958028445678409, 2.21733615874305099104968466724, 2.53390951046190893298063997490, 3.09295236536296196185120877386, 3.77009077410526096369120031381, 4.73651342509810764751574597705, 5.65774866923759257991832112799, 5.72469094873573646097481535324, 6.72161019973575327186699422367, 7.07084190151144290529340672358, 7.62947764982484849351923290378, 8.051520485740927865966210989715, 8.059834272649901480431101846871, 9.260934084989844653679725901204, 9.748363007805241489698153675972, 9.793172155439180332390666150209, 10.48829488288820188849849077179, 11.25660373742175658794531957095, 11.46955939394818327745110718453, 11.71713060033849522735567827264
