| L(s) = 1 | + 3·3-s + 10·5-s − 7-s − 20·9-s + 27·11-s − 15·13-s + 30·15-s − 79·17-s + 71·19-s − 3·21-s + 46·23-s + 75·25-s − 66·27-s − 430·29-s + 305·31-s + 81·33-s − 10·35-s − 68·37-s − 45·39-s − 593·41-s − 648·43-s − 200·45-s − 382·47-s − 4·49-s − 237·51-s − 464·53-s + 270·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.0539·7-s − 0.740·9-s + 0.740·11-s − 0.320·13-s + 0.516·15-s − 1.12·17-s + 0.857·19-s − 0.0311·21-s + 0.417·23-s + 3/5·25-s − 0.470·27-s − 2.75·29-s + 1.76·31-s + 0.427·33-s − 0.0482·35-s − 0.302·37-s − 0.184·39-s − 2.25·41-s − 2.29·43-s − 0.662·45-s − 1.18·47-s − 0.0116·49-s − 0.650·51-s − 1.20·53-s + 0.661·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 29 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 5 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 27 T + 2163 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 15 T + 4205 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 79 T + 10051 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 71 T + 10373 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 430 T + 3182 p T^{2} + 430 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 305 T + 74963 T^{2} - 305 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 68 T + 81098 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 593 T + 222457 T^{2} + 593 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 648 T + 262246 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 382 T + 94906 T^{2} + 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 464 T + 298822 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 331378 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 365439 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 60 T + 445030 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1029 T + 724137 T^{2} - 1029 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 74 T + 674654 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 692 T + 299630 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1460 T + 1111418 T^{2} - 1460 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 220 T + 539138 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1339 T + 1150631 T^{2} - 1339 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.701322903433158067702732273486, −8.446664525328461805884287895393, −7.83477707103550026783754075784, −7.77857940557318661660860604161, −6.82948420685838525052114826048, −6.75256284374257179793625771287, −6.44987532793061732214804910705, −5.96227740284524072450800879758, −5.24337541215581163018212446228, −5.13947727604961232461098502731, −4.79644507135832518064647568702, −3.98812111668178974093637321940, −3.41937363141524061153386990619, −3.33144155729714791004840405271, −2.57318259802550318513414176976, −2.21306781400563819729366918334, −1.56219620301046376305950213672, −1.29833349862638906188854664506, 0, 0,
1.29833349862638906188854664506, 1.56219620301046376305950213672, 2.21306781400563819729366918334, 2.57318259802550318513414176976, 3.33144155729714791004840405271, 3.41937363141524061153386990619, 3.98812111668178974093637321940, 4.79644507135832518064647568702, 5.13947727604961232461098502731, 5.24337541215581163018212446228, 5.96227740284524072450800879758, 6.44987532793061732214804910705, 6.75256284374257179793625771287, 6.82948420685838525052114826048, 7.77857940557318661660860604161, 7.83477707103550026783754075784, 8.446664525328461805884287895393, 8.701322903433158067702732273486
