| L(s) = 1 | − 2·2-s + 4·3-s + 2·4-s − 8·6-s + 8·9-s − 16·11-s + 8·12-s − 6·13-s − 4·16-s − 14·17-s − 16·18-s + 32·22-s − 4·23-s − 25·25-s + 12·26-s + 36·27-s − 104·31-s + 8·32-s − 64·33-s + 28·34-s + 16·36-s − 6·37-s − 24·39-s + 16·41-s − 84·43-s − 32·44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 2-s + 4/3·3-s + 1/2·4-s − 4/3·6-s + 8/9·9-s − 1.45·11-s + 2/3·12-s − 0.461·13-s − 1/4·16-s − 0.823·17-s − 8/9·18-s + 1.45·22-s − 0.173·23-s − 25-s + 6/13·26-s + 4/3·27-s − 3.35·31-s + 1/4·32-s − 1.93·33-s + 0.823·34-s + 4/9·36-s − 0.162·37-s − 0.615·39-s + 0.390·41-s − 1.95·43-s − 0.727·44-s + 4/23·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.003383739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003383739\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 84 T + 3528 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 124 T + 7688 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 94 T + 4418 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9442 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.81723544830392380277792708597, −10.20472395532578646125634402473, −10.18098052018954231364600002650, −9.477031595699162256955455371676, −9.141268292069782617519987736448, −8.776120356321155462373302008104, −8.157451646993707909230284565480, −8.155174804634571168044242753486, −7.45198018515952107404748905383, −7.09695541464832243307087124796, −6.76844036089540480799115309114, −5.78203109036303042538656275326, −5.31053474825517591672987491799, −4.85563381940162307957273698765, −3.84380557111556049011140733996, −3.66002745272228595442641070965, −2.66465637967517539330946638020, −2.27875040427668673378062811026, −1.77701252894254841761466145373, −0.41547412212647844524354877098,
0.41547412212647844524354877098, 1.77701252894254841761466145373, 2.27875040427668673378062811026, 2.66465637967517539330946638020, 3.66002745272228595442641070965, 3.84380557111556049011140733996, 4.85563381940162307957273698765, 5.31053474825517591672987491799, 5.78203109036303042538656275326, 6.76844036089540480799115309114, 7.09695541464832243307087124796, 7.45198018515952107404748905383, 8.155174804634571168044242753486, 8.157451646993707909230284565480, 8.776120356321155462373302008104, 9.141268292069782617519987736448, 9.477031595699162256955455371676, 10.18098052018954231364600002650, 10.20472395532578646125634402473, 10.81723544830392380277792708597
