| L(s) = 1 | − 234·2-s − 2.18e3·3-s + 2.19e4·4-s + 2.80e5·5-s + 5.11e5·6-s − 1.37e6·7-s + 2.52e6·8-s + 4.78e6·9-s − 6.56e7·10-s + 3.40e7·11-s − 4.80e7·12-s + 3.84e8·13-s + 3.21e8·14-s − 6.13e8·15-s − 1.31e9·16-s + 1.25e9·17-s − 1.11e9·18-s − 2.49e9·19-s + 6.17e9·20-s + 3.00e9·21-s − 7.96e9·22-s + 1.12e10·23-s − 5.51e9·24-s + 4.82e10·25-s − 8.98e10·26-s − 1.04e10·27-s − 3.01e10·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 0.577·3-s + 0.671·4-s + 1.60·5-s + 0.746·6-s − 0.630·7-s + 0.425·8-s + 1/3·9-s − 2.07·10-s + 0.526·11-s − 0.387·12-s + 1.69·13-s + 0.814·14-s − 0.927·15-s − 1.22·16-s + 0.744·17-s − 0.430·18-s − 0.641·19-s + 1.07·20-s + 0.363·21-s − 0.680·22-s + 0.691·23-s − 0.245·24-s + 1.58·25-s − 2.19·26-s − 0.192·27-s − 0.422·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.8302241191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8302241191\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{7} T \) |
| good | 2 | \( 1 + 117 p T + p^{15} T^{2} \) |
| 5 | \( 1 - 56142 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 196192 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 3093732 p T + p^{15} T^{2} \) |
| 13 | \( 1 - 29540174 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 1259207586 T + p^{15} T^{2} \) |
| 19 | \( 1 + 2499071020 T + p^{15} T^{2} \) |
| 23 | \( 1 - 11284833672 T + p^{15} T^{2} \) |
| 29 | \( 1 + 48413458530 T + p^{15} T^{2} \) |
| 31 | \( 1 - 130547265752 T + p^{15} T^{2} \) |
| 37 | \( 1 + 200223317554 T + p^{15} T^{2} \) |
| 41 | \( 1 - 679141724202 T + p^{15} T^{2} \) |
| 43 | \( 1 - 279482194892 T + p^{15} T^{2} \) |
| 47 | \( 1 - 1520672832576 T + p^{15} T^{2} \) |
| 53 | \( 1 - 2646053822502 T + p^{15} T^{2} \) |
| 59 | \( 1 - 7399371294540 T + p^{15} T^{2} \) |
| 61 | \( 1 + 42659617819498 T + p^{15} T^{2} \) |
| 67 | \( 1 + 56408026065964 T + p^{15} T^{2} \) |
| 71 | \( 1 + 133149677299848 T + p^{15} T^{2} \) |
| 73 | \( 1 - 105603350884922 T + p^{15} T^{2} \) |
| 79 | \( 1 + 55665674361880 T + p^{15} T^{2} \) |
| 83 | \( 1 - 378077412997332 T + p^{15} T^{2} \) |
| 89 | \( 1 - 219315065897610 T + p^{15} T^{2} \) |
| 97 | \( 1 - 703322682162626 T + p^{15} T^{2} \) |
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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
−22.75854187837434236960520767148, −21.05095180809462034099014819751, −18.84097058576631269679983164911, −17.62349017132223260427771575569, −16.51952975403141132123593654022, −13.42935403633183805303597813533, −10.54110150037755297123906430480, −9.165549554157944895276169478876, −6.24480447745365124568462919800, −1.28064911004270418350661674346,
1.28064911004270418350661674346, 6.24480447745365124568462919800, 9.165549554157944895276169478876, 10.54110150037755297123906430480, 13.42935403633183805303597813533, 16.51952975403141132123593654022, 17.62349017132223260427771575569, 18.84097058576631269679983164911, 21.05095180809462034099014819751, 22.75854187837434236960520767148
