| L(s) = 1 | + 69·3-s − 174·7-s + 2.57e3·9-s + 7.11e3·11-s + 468·13-s + 9.55e3·17-s − 4.26e4·19-s − 1.20e4·21-s + 7.75e4·23-s + 2.67e4·27-s − 6.13e4·29-s + 2.51e5·31-s + 4.90e5·33-s + 8.34e4·37-s + 3.22e4·39-s + 3.63e5·41-s + 3.41e4·43-s + 7.08e5·47-s − 7.93e5·49-s + 6.59e5·51-s + 8.91e5·53-s − 2.93e6·57-s + 2.80e6·59-s − 3.21e6·61-s − 4.47e5·63-s − 1.37e6·67-s + 5.34e6·69-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.191·7-s + 1.17·9-s + 1.61·11-s + 0.0590·13-s + 0.471·17-s − 1.42·19-s − 0.282·21-s + 1.32·23-s + 0.261·27-s − 0.466·29-s + 1.51·31-s + 2.37·33-s + 0.270·37-s + 0.0871·39-s + 0.823·41-s + 0.0655·43-s + 0.995·47-s − 0.963·49-s + 0.695·51-s + 0.822·53-s − 2.10·57-s + 1.78·59-s − 1.81·61-s − 0.225·63-s − 0.557·67-s + 1.96·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.147168207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.147168207\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 23 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 174 T + p^{7} T^{2} \) |
| 11 | \( 1 - 7111 T + p^{7} T^{2} \) |
| 13 | \( 1 - 36 p T + p^{7} T^{2} \) |
| 17 | \( 1 - 9555 T + p^{7} T^{2} \) |
| 19 | \( 1 + 42601 T + p^{7} T^{2} \) |
| 23 | \( 1 - 77526 T + p^{7} T^{2} \) |
| 29 | \( 1 + 61312 T + p^{7} T^{2} \) |
| 31 | \( 1 - 251710 T + p^{7} T^{2} \) |
| 37 | \( 1 - 83462 T + p^{7} T^{2} \) |
| 41 | \( 1 - 363477 T + p^{7} T^{2} \) |
| 43 | \( 1 - 34188 T + p^{7} T^{2} \) |
| 47 | \( 1 - 708812 T + p^{7} T^{2} \) |
| 53 | \( 1 - 891762 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2809152 T + p^{7} T^{2} \) |
| 61 | \( 1 + 3211510 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1372033 T + p^{7} T^{2} \) |
| 71 | \( 1 - 4508308 T + p^{7} T^{2} \) |
| 73 | \( 1 + 628179 T + p^{7} T^{2} \) |
| 79 | \( 1 - 6130474 T + p^{7} T^{2} \) |
| 83 | \( 1 + 9921981 T + p^{7} T^{2} \) |
| 89 | \( 1 - 1806599 T + p^{7} T^{2} \) |
| 97 | \( 1 + 11676482 T + p^{7} T^{2} \) |
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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
−11.14871729964829133129510440248, −9.866596046456988818755986768251, −9.039990701363541108747315157096, −8.425822955599581930428040179437, −7.23483959844173527970931189534, −6.22769979597744438789903169300, −4.39450799148665595811187862923, −3.48331598556110100812024700039, −2.36269973482364577566660554879, −1.09316959064928042871523792270,
1.09316959064928042871523792270, 2.36269973482364577566660554879, 3.48331598556110100812024700039, 4.39450799148665595811187862923, 6.22769979597744438789903169300, 7.23483959844173527970931189534, 8.425822955599581930428040179437, 9.039990701363541108747315157096, 9.866596046456988818755986768251, 11.14871729964829133129510440248
