| L(s) = 1 | − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 11-s − 3·13-s + 4·14-s + 5·16-s + 17-s + 4·19-s − 2·22-s − 23-s + 6·26-s − 6·28-s + 6·29-s − 6·32-s − 2·34-s − 9·37-s − 8·38-s + 5·41-s − 5·43-s + 3·44-s + 2·46-s + 8·47-s + 3·49-s − 9·52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 0.301·11-s − 0.832·13-s + 1.06·14-s + 5/4·16-s + 0.242·17-s + 0.917·19-s − 0.426·22-s − 0.208·23-s + 1.17·26-s − 1.13·28-s + 1.11·29-s − 1.06·32-s − 0.342·34-s − 1.47·37-s − 1.29·38-s + 0.780·41-s − 0.762·43-s + 0.452·44-s + 0.294·46-s + 1.16·47-s + 3/7·49-s − 1.24·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9163257365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9163257365\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 37 | $C_4$ | \( 1 + 9 T + 76 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 70 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 106 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 130 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 238 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 146 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−7.71908043382278030763060421986, −7.48836949510717056080866912172, −7.29234002491812494533287666926, −7.09596453708256747839584957695, −6.57469348446507280620162358264, −6.23013564958981519251997305178, −6.07262738367542801885064891195, −5.57042903265580704764282812157, −5.10088366370866449902772319642, −4.99436815572229374098429029925, −4.26120280453599455818969157367, −3.95285473422983661794824710151, −3.48269952279856671169926783436, −3.09269961637764938544372603644, −2.59805435323362493869224189093, −2.53822204602065502074253016022, −1.75133663519928658571096729052, −1.43041048313580997547658532848, −0.812285980187168075553378598436, −0.35497189319626358947250001610,
0.35497189319626358947250001610, 0.812285980187168075553378598436, 1.43041048313580997547658532848, 1.75133663519928658571096729052, 2.53822204602065502074253016022, 2.59805435323362493869224189093, 3.09269961637764938544372603644, 3.48269952279856671169926783436, 3.95285473422983661794824710151, 4.26120280453599455818969157367, 4.99436815572229374098429029925, 5.10088366370866449902772319642, 5.57042903265580704764282812157, 6.07262738367542801885064891195, 6.23013564958981519251997305178, 6.57469348446507280620162358264, 7.09596453708256747839584957695, 7.29234002491812494533287666926, 7.48836949510717056080866912172, 7.71908043382278030763060421986
