| 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
| \(L(2)\) |
\(\approx\) |
\(4.319219725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.319219725\) |
| \(L(\frac{5}{2})\) |
|
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
−12.19019891942224951465890447375, −11.58243418425407907771342564189, −11.30496087993865054132007647148, −10.98847634051253087292817252885, −10.29035985607098953608329772925, −9.590891518305102290988432101320, −9.146723365077019348941730067500, −8.820297088352843474354050173484, −8.119620680281278902308705138260, −8.115639746554935950875759472259, −7.01345754005955922032390031789, −6.72916188792329367570159562389, −5.82681535337982529505919996246, −5.60135589540183461344715700402, −4.66596494983546019280649525305, −3.85447943702866990821757377403, −3.21432629344216440455168142113, −2.94627604639864030210588474962, −1.44957713415545444810921398325, −1.03604333643836670549820646941,
1.03604333643836670549820646941, 1.44957713415545444810921398325, 2.94627604639864030210588474962, 3.21432629344216440455168142113, 3.85447943702866990821757377403, 4.66596494983546019280649525305, 5.60135589540183461344715700402, 5.82681535337982529505919996246, 6.72916188792329367570159562389, 7.01345754005955922032390031789, 8.115639746554935950875759472259, 8.119620680281278902308705138260, 8.820297088352843474354050173484, 9.146723365077019348941730067500, 9.590891518305102290988432101320, 10.29035985607098953608329772925, 10.98847634051253087292817252885, 11.30496087993865054132007647148, 11.58243418425407907771342564189, 12.19019891942224951465890447375
