| L(s) = 1 | + 11·3-s − 25·5-s − 49·7-s − 122·9-s + 267·11-s − 1.08e3·13-s − 275·15-s − 513·17-s + 802·19-s − 539·21-s + 1.29e3·23-s + 625·25-s − 4.01e3·27-s + 1.77e3·29-s + 2.58e3·31-s + 2.93e3·33-s + 1.22e3·35-s + 1.38e4·37-s − 1.19e4·39-s − 1.19e4·41-s + 598·43-s + 3.05e3·45-s + 1.70e4·47-s + 2.40e3·49-s − 5.64e3·51-s + 2.78e4·53-s − 6.67e3·55-s + ⋯ |
| L(s) = 1 | + 0.705·3-s − 0.447·5-s − 0.377·7-s − 0.502·9-s + 0.665·11-s − 1.78·13-s − 0.315·15-s − 0.430·17-s + 0.509·19-s − 0.266·21-s + 0.508·23-s + 1/5·25-s − 1.05·27-s + 0.392·29-s + 0.482·31-s + 0.469·33-s + 0.169·35-s + 1.66·37-s − 1.25·39-s − 1.10·41-s + 0.0493·43-s + 0.224·45-s + 1.12·47-s + 1/7·49-s − 0.303·51-s + 1.36·53-s − 0.297·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.840607616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840607616\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 11 T + p^{5} T^{2} \) |
| 11 | \( 1 - 267 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1087 T + p^{5} T^{2} \) |
| 17 | \( 1 + 513 T + p^{5} T^{2} \) |
| 19 | \( 1 - 802 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1290 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1779 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2584 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13862 T + p^{5} T^{2} \) |
| 41 | \( 1 + 11904 T + p^{5} T^{2} \) |
| 43 | \( 1 - 598 T + p^{5} T^{2} \) |
| 47 | \( 1 - 17019 T + p^{5} T^{2} \) |
| 53 | \( 1 - 27852 T + p^{5} T^{2} \) |
| 59 | \( 1 + 30912 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1780 T + p^{5} T^{2} \) |
| 67 | \( 1 + 25052 T + p^{5} T^{2} \) |
| 71 | \( 1 - 51984 T + p^{5} T^{2} \) |
| 73 | \( 1 - 47690 T + p^{5} T^{2} \) |
| 79 | \( 1 - 102121 T + p^{5} T^{2} \) |
| 83 | \( 1 - 83676 T + p^{5} T^{2} \) |
| 89 | \( 1 + 32400 T + p^{5} T^{2} \) |
| 97 | \( 1 + 148645 T + p^{5} T^{2} \) |
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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
−9.685217489136929876532401482308, −9.203772050057122624776058999023, −8.197803174520189394061605679034, −7.40376269087334489308447048222, −6.52798668033269083030857023849, −5.24683370626312080338183158694, −4.20857848538291660616281787201, −3.08719208046657794689392510134, −2.30620156561801784641769181084, −0.62044504649449934031384600739,
0.62044504649449934031384600739, 2.30620156561801784641769181084, 3.08719208046657794689392510134, 4.20857848538291660616281787201, 5.24683370626312080338183158694, 6.52798668033269083030857023849, 7.40376269087334489308447048222, 8.197803174520189394061605679034, 9.203772050057122624776058999023, 9.685217489136929876532401482308
