| L(s) = 1 | − 3-s − 7·5-s − 26·9-s − 35·11-s − 66·13-s + 7·15-s − 59·17-s + 137·19-s + 7·23-s − 76·25-s + 53·27-s + 106·29-s + 75·31-s + 35·33-s + 11·37-s + 66·39-s + 498·41-s − 260·43-s + 182·45-s − 171·47-s + 59·51-s − 417·53-s + 245·55-s − 137·57-s − 17·59-s − 51·61-s + 462·65-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.626·5-s − 0.962·9-s − 0.959·11-s − 1.40·13-s + 0.120·15-s − 0.841·17-s + 1.65·19-s + 0.0634·23-s − 0.607·25-s + 0.377·27-s + 0.678·29-s + 0.434·31-s + 0.184·33-s + 0.0488·37-s + 0.270·39-s + 1.89·41-s − 0.922·43-s + 0.602·45-s − 0.530·47-s + 0.161·51-s − 1.08·53-s + 0.600·55-s − 0.318·57-s − 0.0375·59-s − 0.107·61-s + 0.881·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8119021882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8119021882\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 35 T + p^{3} T^{2} \) |
| 13 | \( 1 + 66 T + p^{3} T^{2} \) |
| 17 | \( 1 + 59 T + p^{3} T^{2} \) |
| 19 | \( 1 - 137 T + p^{3} T^{2} \) |
| 23 | \( 1 - 7 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 75 T + p^{3} T^{2} \) |
| 37 | \( 1 - 11 T + p^{3} T^{2} \) |
| 41 | \( 1 - 498 T + p^{3} T^{2} \) |
| 43 | \( 1 + 260 T + p^{3} T^{2} \) |
| 47 | \( 1 + 171 T + p^{3} T^{2} \) |
| 53 | \( 1 + 417 T + p^{3} T^{2} \) |
| 59 | \( 1 + 17 T + p^{3} T^{2} \) |
| 61 | \( 1 + 51 T + p^{3} T^{2} \) |
| 67 | \( 1 + 439 T + p^{3} T^{2} \) |
| 71 | \( 1 - 784 T + p^{3} T^{2} \) |
| 73 | \( 1 + 295 T + p^{3} T^{2} \) |
| 79 | \( 1 - 495 T + p^{3} T^{2} \) |
| 83 | \( 1 - 932 T + p^{3} T^{2} \) |
| 89 | \( 1 - 873 T + p^{3} T^{2} \) |
| 97 | \( 1 - 290 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.890807752383550103525542908105, −9.100844916939944434208265611783, −7.944491169504093428405659828483, −7.57054450107503179303590282682, −6.39319440952847158196571616336, −5.29916290740414099181049918632, −4.66380340167695564945054129483, −3.23699757063132437972640999871, −2.39507779295500742235830226430, −0.48247720709813489244568734978,
0.48247720709813489244568734978, 2.39507779295500742235830226430, 3.23699757063132437972640999871, 4.66380340167695564945054129483, 5.29916290740414099181049918632, 6.39319440952847158196571616336, 7.57054450107503179303590282682, 7.944491169504093428405659828483, 9.100844916939944434208265611783, 9.890807752383550103525542908105
