| L(s) = 1 | − 2·3-s − 2·4-s − 15·9-s + 14·11-s + 4·12-s + 4·16-s − 34·23-s + 50·27-s + 34·31-s − 28·33-s + 30·36-s − 94·37-s − 28·44-s + 116·47-s − 8·48-s + 26·49-s − 4·53-s − 110·59-s − 8·64-s − 178·67-s + 68·69-s − 14·71-s + 140·81-s − 194·89-s + 68·92-s − 68·93-s + 242·97-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 1/2·4-s − 5/3·9-s + 1.27·11-s + 1/3·12-s + 1/4·16-s − 1.47·23-s + 1.85·27-s + 1.09·31-s − 0.848·33-s + 5/6·36-s − 2.54·37-s − 0.636·44-s + 2.46·47-s − 1/6·48-s + 0.530·49-s − 0.0754·53-s − 1.86·59-s − 1/8·64-s − 2.65·67-s + 0.985·69-s − 0.197·71-s + 1.72·81-s − 2.17·89-s + 0.739·92-s − 0.731·93-s + 2.49·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6387421058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6387421058\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 14 T + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 26 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 266 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 70 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 74 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3410 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 242 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 89 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 5542 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11330 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12626 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 121 T + p^{2} T^{2} )^{2} \) |
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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
−10.73406216707013856330998945468, −10.29844631800142809338170271274, −10.21147180372204068866510122431, −9.279868907307245706834166775879, −8.871855166252757322571568332013, −8.853403449351063063507526591558, −8.285844285297308456968477553250, −7.66416152340962625399428021614, −7.24154147383847849549567517273, −6.39668900968460893681824980723, −6.28112589983982775409990639499, −5.69318296671146651417641921757, −5.42034299918719778459836369559, −4.63585754970875912102820291767, −4.25619744154938485020322118544, −3.53436877559430937119423676310, −3.05009044123995312551541275156, −2.23851789080031646694036441241, −1.32797959855306721159989798168, −0.34380741430907695462233130714,
0.34380741430907695462233130714, 1.32797959855306721159989798168, 2.23851789080031646694036441241, 3.05009044123995312551541275156, 3.53436877559430937119423676310, 4.25619744154938485020322118544, 4.63585754970875912102820291767, 5.42034299918719778459836369559, 5.69318296671146651417641921757, 6.28112589983982775409990639499, 6.39668900968460893681824980723, 7.24154147383847849549567517273, 7.66416152340962625399428021614, 8.285844285297308456968477553250, 8.853403449351063063507526591558, 8.871855166252757322571568332013, 9.279868907307245706834166775879, 10.21147180372204068866510122431, 10.29844631800142809338170271274, 10.73406216707013856330998945468
