| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s + 3·9-s − 10·11-s − 2·14-s + 5·16-s − 6·18-s + 20·22-s + 3·28-s − 12·29-s − 6·32-s + 9·36-s − 16·37-s + 16·43-s − 30·44-s + 49-s − 8·53-s − 4·56-s + 24·58-s + 3·63-s + 7·64-s − 18·67-s − 20·71-s − 12·72-s + 32·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s + 9-s − 3.01·11-s − 0.534·14-s + 5/4·16-s − 1.41·18-s + 4.26·22-s + 0.566·28-s − 2.22·29-s − 1.06·32-s + 3/2·36-s − 2.63·37-s + 2.43·43-s − 4.52·44-s + 1/7·49-s − 1.09·53-s − 0.534·56-s + 3.15·58-s + 0.377·63-s + 7/8·64-s − 2.19·67-s − 2.37·71-s − 1.41·72-s + 3.71·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3763793203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3763793203\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.982704647601194275162559588174, −7.909616856504655698109693901245, −7.35159550652469145874551234062, −7.28202966725007180028492050863, −6.75922904616944609353713204114, −5.74372284090584252527668413358, −5.69551226567491875356313493124, −5.23317326717424642267182154953, −4.53392433226581882017134096766, −4.02569848626692601267512154263, −2.96910811860602576493917153273, −2.90932005195183109143742073385, −1.82578348423128801401226534071, −1.76448136119712918970498775230, −0.34762960201747218551587326811,
0.34762960201747218551587326811, 1.76448136119712918970498775230, 1.82578348423128801401226534071, 2.90932005195183109143742073385, 2.96910811860602576493917153273, 4.02569848626692601267512154263, 4.53392433226581882017134096766, 5.23317326717424642267182154953, 5.69551226567491875356313493124, 5.74372284090584252527668413358, 6.75922904616944609353713204114, 7.28202966725007180028492050863, 7.35159550652469145874551234062, 7.909616856504655698109693901245, 7.982704647601194275162559588174
