| L(s) = 1 | + 2·3-s + 5·5-s − 28·7-s + 27·9-s − 27·11-s + 82·13-s + 10·15-s − 6·17-s + 49·19-s − 56·21-s + 81·23-s + 154·27-s + 132·29-s − 188·31-s − 54·33-s − 140·35-s − 23·37-s + 164·39-s + 102·41-s − 404·43-s + 135·45-s − 327·47-s + 441·49-s − 12·51-s + 201·53-s − 135·55-s + 98·57-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.447·5-s − 1.51·7-s + 9-s − 0.740·11-s + 1.74·13-s + 0.172·15-s − 0.0856·17-s + 0.591·19-s − 0.581·21-s + 0.734·23-s + 1.09·27-s + 0.845·29-s − 1.08·31-s − 0.284·33-s − 0.676·35-s − 0.102·37-s + 0.673·39-s + 0.388·41-s − 1.43·43-s + 0.447·45-s − 1.01·47-s + 9/7·49-s − 0.0329·51-s + 0.520·53-s − 0.330·55-s + 0.227·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.503045988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.503045988\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T - 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 27 T - 602 T^{2} + 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 41 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T - 4877 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 49 T - 4458 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 81 T - 5606 T^{2} - 81 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 188 T + 5553 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 23 T - 50124 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 51 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 202 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 327 T + 3106 T^{2} + 327 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 201 T - 108476 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 24 T - 204803 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 196 T - 188565 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 964 T + 628533 T^{2} - 964 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1140 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 868 T + 364407 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 638 T - 85995 T^{2} + 638 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 810 T - 48869 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1550 T + p^{3} T^{2} )^{2} \) |
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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
−13.00646404908557780691654491806, −12.77632585149470054290027553486, −12.01849635359060297523189658920, −11.38565444595398364837814196245, −10.71118197280258131875364475956, −10.31552709278002514521781202272, −9.851700930834103030008865907277, −9.419066162793880561514314105027, −8.673154028063971641925109562582, −8.475050632117656949452374280092, −7.47651489351239943825845568356, −6.99717429755133479808586364163, −6.42764282738583235764828602744, −5.93288474942492407743579392837, −5.16229729877004183891882443792, −4.32816138162602027768394727374, −3.33743685862819342712662381318, −3.16141328043408152311203677691, −1.89027191285234362083628203255, −0.805483017785080277838015410860,
0.805483017785080277838015410860, 1.89027191285234362083628203255, 3.16141328043408152311203677691, 3.33743685862819342712662381318, 4.32816138162602027768394727374, 5.16229729877004183891882443792, 5.93288474942492407743579392837, 6.42764282738583235764828602744, 6.99717429755133479808586364163, 7.47651489351239943825845568356, 8.475050632117656949452374280092, 8.673154028063971641925109562582, 9.419066162793880561514314105027, 9.851700930834103030008865907277, 10.31552709278002514521781202272, 10.71118197280258131875364475956, 11.38565444595398364837814196245, 12.01849635359060297523189658920, 12.77632585149470054290027553486, 13.00646404908557780691654491806
