| L(s) = 1 | − 4·3-s − 4·7-s − 18·9-s + 40·11-s − 104·13-s − 96·17-s + 40·19-s + 16·21-s − 284·23-s + 100·27-s − 140·29-s + 192·31-s − 160·33-s − 200·37-s + 416·39-s − 524·41-s − 372·43-s − 84·47-s − 458·49-s + 384·51-s + 296·53-s − 160·57-s + 696·59-s − 692·61-s + 72·63-s − 316·67-s + 1.13e3·69-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.215·7-s − 2/3·9-s + 1.09·11-s − 2.21·13-s − 1.36·17-s + 0.482·19-s + 0.166·21-s − 2.57·23-s + 0.712·27-s − 0.896·29-s + 1.11·31-s − 0.844·33-s − 0.888·37-s + 1.70·39-s − 1.99·41-s − 1.31·43-s − 0.260·47-s − 1.33·49-s + 1.05·51-s + 0.767·53-s − 0.371·57-s + 1.53·59-s − 1.45·61-s + 0.143·63-s − 0.576·67-s + 1.98·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 474 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 40 T + 1526 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 p T + 7002 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 96 T + 5986 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 40 T + 12582 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 284 T + 40442 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 140 T + 47534 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 192 T + 13502 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 200 T + 83562 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 524 T + 203030 T^{2} + 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 372 T + 183026 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 84 T + 108010 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 296 T + 29258 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 696 T + 346006 T^{2} - 696 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 692 T + 554862 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 316 T + 625314 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 688 T + 809582 T^{2} - 688 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 656 T + 682482 T^{2} - 656 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 736 T + 1115358 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1628 T + 1748546 T^{2} - 1628 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 660 T + 1463542 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 896 T + 1987650 T^{2} + 896 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.77701039166857319628550511288, −11.53829750754921530882300939385, −10.88586272866107833600286953432, −10.17119908318087921660469584465, −9.710627189470849973571592719041, −9.641824678457640394295994850163, −8.575981530445911839527308402912, −8.437820888546498332616448012185, −7.57308712480603523737742873798, −7.05469310000021774118208615817, −6.30259506404545649370071310088, −6.29688782840226073788380022922, −5.09652436693813676930357903579, −5.08644831579149264954657335601, −4.13272190263296367101474251816, −3.45476739735398919789640093568, −2.42335435776806323719102453894, −1.79036262265578200098059158813, 0, 0,
1.79036262265578200098059158813, 2.42335435776806323719102453894, 3.45476739735398919789640093568, 4.13272190263296367101474251816, 5.08644831579149264954657335601, 5.09652436693813676930357903579, 6.29688782840226073788380022922, 6.30259506404545649370071310088, 7.05469310000021774118208615817, 7.57308712480603523737742873798, 8.437820888546498332616448012185, 8.575981530445911839527308402912, 9.641824678457640394295994850163, 9.710627189470849973571592719041, 10.17119908318087921660469584465, 10.88586272866107833600286953432, 11.53829750754921530882300939385, 11.77701039166857319628550511288
