| L(s) = 1 | − 2·3-s − 4-s + 4·7-s + 9-s + 2·12-s − 3·16-s − 8·21-s − 25-s + 4·27-s − 4·28-s − 20·31-s − 36-s + 4·43-s + 6·48-s − 2·49-s − 14·61-s + 4·63-s + 7·64-s + 10·73-s + 2·75-s − 11·81-s + 8·84-s + 40·93-s − 14·97-s + 100-s − 4·108-s + 2·109-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.577·12-s − 3/4·16-s − 1.74·21-s − 1/5·25-s + 0.769·27-s − 0.755·28-s − 3.59·31-s − 1/6·36-s + 0.609·43-s + 0.866·48-s − 2/7·49-s − 1.79·61-s + 0.503·63-s + 7/8·64-s + 1.17·73-s + 0.230·75-s − 1.22·81-s + 0.872·84-s + 4.14·93-s − 1.42·97-s + 1/10·100-s − 0.384·108-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 109 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−9.241039402548484915567381296413, −8.868578662855853203898028087059, −8.286920473798172753174354214638, −7.77239562841085857542418185035, −7.29052323070261942343367056724, −6.77873667936168530448599404329, −6.06549369200561161055042121246, −5.53402102397877396607861005811, −5.15489792446163350962030773478, −4.68883587182190550196803989760, −4.14275095283320908899231925739, −3.38861252910552575582491781098, −2.19372602203621639198366327936, −1.46786189455252209443063613482, 0,
1.46786189455252209443063613482, 2.19372602203621639198366327936, 3.38861252910552575582491781098, 4.14275095283320908899231925739, 4.68883587182190550196803989760, 5.15489792446163350962030773478, 5.53402102397877396607861005811, 6.06549369200561161055042121246, 6.77873667936168530448599404329, 7.29052323070261942343367056724, 7.77239562841085857542418185035, 8.286920473798172753174354214638, 8.868578662855853203898028087059, 9.241039402548484915567381296413
