| L(s) = 1 | + 61.9·3-s + 70·5-s + 1.11e3·7-s + 1.65e3·9-s − 7.25e3·11-s + 1.37e4·13-s + 4.33e3·15-s + 1.69e4·17-s + 3.40e4·19-s + 6.91e4·21-s − 3.23e4·23-s − 7.32e4·25-s − 3.30e4·27-s + 3.41e4·29-s − 1.20e5·31-s − 4.49e5·33-s + 7.80e4·35-s + 3.52e4·37-s + 8.52e5·39-s − 4.84e5·41-s − 6.72e5·43-s + 1.15e5·45-s − 1.20e6·47-s + 4.20e5·49-s + 1.05e6·51-s + 8.51e5·53-s − 5.07e5·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 0.250·5-s + 1.22·7-s + 0.755·9-s − 1.64·11-s + 1.73·13-s + 0.331·15-s + 0.838·17-s + 1.13·19-s + 1.62·21-s − 0.554·23-s − 0.937·25-s − 0.323·27-s + 0.260·29-s − 0.726·31-s − 2.17·33-s + 0.307·35-s + 0.114·37-s + 2.30·39-s − 1.09·41-s − 1.28·43-s + 0.189·45-s − 1.69·47-s + 0.510·49-s + 1.11·51-s + 0.785·53-s − 0.411·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.917293203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.917293203\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 61.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 70T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.11e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.25e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.37e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.69e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.23e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.41e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.52e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.84e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.72e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.51e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.95e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.16e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.57e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.91e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.14e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.51e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.00e4T + 8.07e13T^{2} \) |
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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
−15.10460115176463603531950070688, −13.95488558825590140497919544176, −13.32285673699580389525958593877, −11.41798884937890434071286807293, −10.01755121287953035290227069540, −8.407707465671385532909313373398, −7.83185609389227802420385575571, −5.39854234455545785684287278371, −3.37944066532676896011382744824, −1.73936096582412474192260767759,
1.73936096582412474192260767759, 3.37944066532676896011382744824, 5.39854234455545785684287278371, 7.83185609389227802420385575571, 8.407707465671385532909313373398, 10.01755121287953035290227069540, 11.41798884937890434071286807293, 13.32285673699580389525958593877, 13.95488558825590140497919544176, 15.10460115176463603531950070688
